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overflow: hidden; text-overflow: ellipsis; -webkit-line-clamp: 3; -webkit-box-orient: vertical; }</style><div class="col-xs-12 clearfix"><div class="u-floatLeft"><h1 class="PageHeader-title u-m0x u-fs30">Analytic Number Theory</h1><div class="u-tcGrayDark">8,312 Followers</div><div class="u-tcGrayDark u-mt2x">Recent papers in <b>Analytic Number Theory</b></div></div></div></div></div></div><div class="TabbedNavigation"><div class="container"><div class="row"><div class="col-xs-12 clearfix"><ul class="nav u-m0x u-p0x list-inline u-displayFlex"><li class="active"><a href="https://www.academia.edu/Documents/in/Analytic_Number_Theory">Top Papers</a></li><li><a href="https://www.academia.edu/Documents/in/Analytic_Number_Theory/MostCited">Most Cited Papers</a></li><li><a href="https://www.academia.edu/Documents/in/Analytic_Number_Theory/MostDownloaded">Most Downloaded Papers</a></li><li><a href="https://www.academia.edu/Documents/in/Analytic_Number_Theory/MostRecent">Newest Papers</a></li><li><a class="" href="https://www.academia.edu/People/Analytic_Number_Theory">People</a></li></ul></div><style type="text/css">ul.nav{flex-direction:row}@media(max-width: 567px){ul.nav{flex-direction:column}.TabbedNavigation li{max-width:100%}.TabbedNavigation li.active{background-color:var(--background-grey, #dddde2)}.TabbedNavigation li.active:before,.TabbedNavigation li.active:after{display:none}}</style></div></div></div><div class="container"><div class="row"><div class="col-xs-12"><div class="u-displayFlex"><div class="u-flexGrow1"><div class="works"><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_8740633" data-work_id="8740633" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/8740633/Unit%C3%A9s_elliptiques_et_groupes_de_classes">Unités elliptiques et groupes de classes</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest">We study the abelian extensions of a quadratic imaginary field<br />and discuss the analogous statements of theorems of Mazur and<br />Wiles.</div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/8740633" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="84e95e4633ee1e002bb087049149e73b" rel="nofollow" data-download="{"attachment_id":35092605,"asset_id":8740633,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/35092605/download_file?st=MTc0MDg2MzQyNyw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="5118693" href="https://usmba.academia.edu/%D9%85%D8%AD%D9%85%D8%AF%D8%A7%D9%84%D8%B4%D8%B1%D9%82%D8%A7%D9%86%D9%8A%D8%A3%D9%84%D8%AD%D8%B3%D9%86%D9%89">محمد الشرقاني ألحسنى</a><script data-card-contents-for-user="5118693" type="text/json">{"id":5118693,"first_name":"محمد","last_name":"الشرقاني ألحسنى","domain_name":"usmba","page_name":"محمدالشرقانيألحسنى","display_name":"محمد الشرقاني ألحسنى","profile_url":"https://usmba.academia.edu/%D9%85%D8%AD%D9%85%D8%AF%D8%A7%D9%84%D8%B4%D8%B1%D9%82%D8%A7%D9%86%D9%8A%D8%A3%D9%84%D8%AD%D8%B3%D9%86%D9%89?f_ri=302","photo":"https://0.academia-photos.com/5118693/2242957/2623578/s65__._.jpg"}</script></span></span></li><li class="js-paper-rank-work_8740633 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="8740633"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 8740633, container: ".js-paper-rank-work_8740633", }); 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$(".js-view-count[data-work-id=8740633]").text(description); $(".js-view-count-work_8740633").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_8740633").removeClass('hidden') })</script></div></li><li class="InlineList-item u-positionRelative" style="max-width: 250px"><div class="u-positionAbsolute" data-has-card-for-ri-list="8740633"><i class="fa fa-tag InlineList-item-icon u-positionRelative"></i></div><span class="InlineList-item-text u-textTruncate u-pl6x"><a class="InlineList-item-text" data-has-card-for-ri="302" rel="nofollow" href="https://www.academia.edu/Documents/in/Analytic_Number_Theory">Analytic Number Theory</a><script data-card-contents-for-ri="302" type="text/json">{"id":302,"name":"Analytic Number Theory","url":"https://www.academia.edu/Documents/in/Analytic_Number_Theory?f_ri=302","nofollow":true}</script></span></li><script>(function(){ if (false) { new Aedu.ResearchInterestListCard({ el: $('*[data-has-card-for-ri-list=8740633]'), work: {"id":8740633,"title":"Unités elliptiques et groupes de classes","created_at":"2014-10-11T23:52:28.496-07:00","url":"https://www.academia.edu/8740633/Unit%C3%A9s_elliptiques_et_groupes_de_classes?f_ri=302","dom_id":"work_8740633","summary":"We study the abelian extensions of a quadratic imaginary field\nand discuss the analogous statements of theorems of Mazur and\nWiles.","downloadable_attachments":[{"id":35092605,"asset_id":8740633,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":5118693,"first_name":"محمد","last_name":"الشرقاني ألحسنى","domain_name":"usmba","page_name":"محمدالشرقانيألحسنى","display_name":"محمد الشرقاني ألحسنى","profile_url":"https://usmba.academia.edu/%D9%85%D8%AD%D9%85%D8%AF%D8%A7%D9%84%D8%B4%D8%B1%D9%82%D8%A7%D9%86%D9%8A%D8%A3%D9%84%D8%AD%D8%B3%D9%86%D9%89?f_ri=302","photo":"https://0.academia-photos.com/5118693/2242957/2623578/s65__._.jpg"}],"research_interests":[{"id":302,"name":"Analytic Number Theory","url":"https://www.academia.edu/Documents/in/Analytic_Number_Theory?f_ri=302","nofollow":true}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_66149337" data-work_id="66149337" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/66149337/A_Conjecture_of_Ducci_Sequences">A Conjecture of Ducci Sequences</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest"><div class="summarized">In this paper, we argue an arithmetic issue also known as Ducci's 4 or nnumber game. First we set an arbitrary non-negative integer at each vertex of a polygon, and apply the following procedure: at the center of each edge, set the... <a class="more_link u-tcGrayDark u-linkUnstyled" data-container=".work_66149337" data-show=".complete" data-hide=".summarized" data-more-link-behavior="true" href="#">more</a></div><div class="complete hidden">In this paper, we argue an arithmetic issue also known as Ducci's 4 or nnumber game. First we set an arbitrary non-negative integer at each vertex of a polygon, and apply the following procedure: at the center of each edge, set the difference of two integers at both ends of the edge, and connect the centers of adjacent edges: in this way, we obtain a new polygon with non-negative integers at every vertex. We apply the same procedure recursively, until all numbers set around the polygon become zeros. In the case the number N of edges of the polygon is four, it has been claimed that the procedure will always terminate in finite steps, and it is proved by B. Freedman [1]. For an arbitrary N, we know the following facts. 1. If the number N of edges of the polygon is a power of two, the recursive procedure will always terminate in finite steps. 2. Otherwise, there exits some non-negative integers at vertices of the polygon, from which the recursive procedure will never terminate. This time, we discuss the following fact: Let A, B, or C be positive integers on consecutive vertices on polygon A≤C≤B, A≥C≥B, B≤A≤C, or B≥A≥C, the recursive procedure by the binary operation which vertices N are four will always terminate by four steps. In this paper, we prove it and also discuss further general case.</div></div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/66149337" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="e88ea4eaa5d492dfeb856b3adee95822" rel="nofollow" data-download="{"attachment_id":77455232,"asset_id":66149337,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/77455232/download_file?st=MTc0MDg2MzQyNyw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="30920916" href="https://jein.academia.edu/MiztaniEuich">Euich Miztani</a><script data-card-contents-for-user="30920916" type="text/json">{"id":30920916,"first_name":"Euich","last_name":"Miztani","domain_name":"jein","page_name":"MiztaniEuich","display_name":"Euich Miztani","profile_url":"https://jein.academia.edu/MiztaniEuich?f_ri=302","photo":"https://0.academia-photos.com/30920916/9044475/143170552/s65_euich.miztani.jpeg"}</script></span></span></li><li class="js-paper-rank-work_66149337 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="66149337"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 66149337, container: ".js-paper-rank-work_66149337", }); 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First we set an arbitrary non-negative integer at each vertex of a polygon, and apply the following procedure: at the center of each edge, set the difference of two integers at both ends of the edge, and connect the centers of adjacent edges: in this way, we obtain a new polygon with non-negative integers at every vertex. We apply the same procedure recursively, until all numbers set around the polygon become zeros. In the case the number N of edges of the polygon is four, it has been claimed that the procedure will always terminate in finite steps, and it is proved by B. Freedman [1]. For an arbitrary N, we know the following facts. 1. If the number N of edges of the polygon is a power of two, the recursive procedure will always terminate in finite steps. 2. Otherwise, there exits some non-negative integers at vertices of the polygon, from which the recursive procedure will never terminate. This time, we discuss the following fact: Let A, B, or C be positive integers on consecutive vertices on polygon A≤C≤B, A≥C≥B, B≤A≤C, or B≥A≥C, the recursive procedure by the binary operation which vertices N are four will always terminate by four steps. In this paper, we prove it and also discuss further general case.","downloadable_attachments":[{"id":77455232,"asset_id":66149337,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":30920916,"first_name":"Euich","last_name":"Miztani","domain_name":"jein","page_name":"MiztaniEuich","display_name":"Euich Miztani","profile_url":"https://jein.academia.edu/MiztaniEuich?f_ri=302","photo":"https://0.academia-photos.com/30920916/9044475/143170552/s65_euich.miztani.jpeg"}],"research_interests":[{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=302","nofollow":true},{"id":302,"name":"Analytic Number Theory","url":"https://www.academia.edu/Documents/in/Analytic_Number_Theory?f_ri=302","nofollow":true},{"id":2616,"name":"Graph Theory","url":"https://www.academia.edu/Documents/in/Graph_Theory?f_ri=302","nofollow":true},{"id":524840,"name":"Integers","url":"https://www.academia.edu/Documents/in/Integers?f_ri=302","nofollow":true}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_28450417 coauthored" data-work_id="28450417" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/28450417/On_intervals_kn_k_1_n_containing_a_prime_for_all_n_1">On intervals (kn,(k+1)n) containing a prime for all n>1</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest"><div class="summarized">We study values of k for which the interval (kn, (k + 1)n) contains a prime for every n > 1. We prove that the list of such integers k includes k = 1, 2, 3, 5, 9, 14, and no others, at least for k ≤ 50, 000, 000. For every known k of this... <a class="more_link u-tcGrayDark u-linkUnstyled" data-container=".work_28450417" data-show=".complete" data-hide=".summarized" data-more-link-behavior="true" href="#">more</a></div><div class="complete hidden">We study values of k for which the interval (kn, (k + 1)n) contains a prime for every n > 1. We prove that the list of such integers k includes k = 1, 2, 3, 5, 9, 14, and no others, at least for k ≤ 50, 000, 000. For every known k of this list, we give a good upper estimate of the smallest N k (m), such that, if n ≥ N k (m), then the interval (kn, (k +1)n) contains at least m primes.</div></div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/28450417" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="9de24d1bb27e6ca50692ce3f0c7b45b9" rel="nofollow" data-download="{"attachment_id":48799647,"asset_id":28450417,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/48799647/download_file?st=MTc0MDg2MzQyNyw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="53353087" href="https://independent.academia.edu/GreathouseCharles">Charles Greathouse</a><script data-card-contents-for-user="53353087" type="text/json">{"id":53353087,"first_name":"Charles","last_name":"Greathouse","domain_name":"independent","page_name":"GreathouseCharles","display_name":"Charles Greathouse","profile_url":"https://independent.academia.edu/GreathouseCharles?f_ri=302","photo":"https://0.academia-photos.com/53353087/14114825/15116986/s65_charles.greathouse.jpg"}</script></span></span><span class="u-displayInlineBlock InlineList-item-text"> and <span class="u-textDecorationUnderline u-clickable InlineList-item-text js-work-more-authors-28450417">+1</span><div class="hidden js-additional-users-28450417"><div><span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a href="https://independent.academia.edu/VShevelev">Vladimir Shevelev</a></span></div></div></span><script>(function(){ var popoverSettings = { el: $('.js-work-more-authors-28450417'), placement: 'bottom', hide_delay: 200, html: true, content: function(){ return $('.js-additional-users-28450417').html(); } } new HoverPopover(popoverSettings); })();</script></li><li class="js-paper-rank-work_28450417 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="28450417"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 28450417, container: ".js-paper-rank-work_28450417", }); });</script></li><li class="js-percentile-work_28450417 InlineList-item InlineList-item--bordered hidden u-tcGrayDark"><span class="percentile-widget hidden"><span class="u-mr2x percentile-widget" style="display: none">•</span><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 28450417; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-percentile-work_28450417"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></li><li class="js-view-count-work_28450417 InlineList-item InlineList-item--bordered hidden"><div><span><span class="js-view-count view-count u-mr2x" data-work-id="28450417"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 28450417; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=28450417]").text(description); $(".js-view-count-work_28450417").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_28450417").removeClass('hidden') })</script></div></li><li class="InlineList-item u-positionRelative" style="max-width: 250px"><div class="u-positionAbsolute" data-has-card-for-ri-list="28450417"><i class="fa fa-tag InlineList-item-icon u-positionRelative"></i>  <a class="InlineList-item-text u-positionRelative">2</a>  </div><span class="InlineList-item-text u-textTruncate u-pl9x"><a class="InlineList-item-text" data-has-card-for-ri="302" rel="nofollow" href="https://www.academia.edu/Documents/in/Analytic_Number_Theory">Analytic Number Theory</a>, <script data-card-contents-for-ri="302" type="text/json">{"id":302,"name":"Analytic Number Theory","url":"https://www.academia.edu/Documents/in/Analytic_Number_Theory?f_ri=302","nofollow":true}</script><a class="InlineList-item-text" data-has-card-for-ri="72443" rel="nofollow" href="https://www.academia.edu/Documents/in/Prime_Numbers">Prime Numbers</a><script data-card-contents-for-ri="72443" type="text/json">{"id":72443,"name":"Prime Numbers","url":"https://www.academia.edu/Documents/in/Prime_Numbers?f_ri=302","nofollow":true}</script></span></li><script>(function(){ if (true) { new Aedu.ResearchInterestListCard({ el: $('*[data-has-card-for-ri-list=28450417]'), work: {"id":28450417,"title":"On intervals (kn,(k+1)n) containing a prime for all n\u003e1","created_at":"2016-09-13T06:22:40.604-07:00","url":"https://www.academia.edu/28450417/On_intervals_kn_k_1_n_containing_a_prime_for_all_n_1?f_ri=302","dom_id":"work_28450417","summary":"We study values of k for which the interval (kn, (k + 1)n) contains a prime for every n \u003e 1. We prove that the list of such integers k includes k = 1, 2, 3, 5, 9, 14, and no others, at least for k ≤ 50, 000, 000. For every known k of this list, we give a good upper estimate of the smallest N k (m), such that, if n ≥ N k (m), then the interval (kn, (k +1)n) contains at least m primes.","downloadable_attachments":[{"id":48799647,"asset_id":28450417,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":53353087,"first_name":"Charles","last_name":"Greathouse","domain_name":"independent","page_name":"GreathouseCharles","display_name":"Charles Greathouse","profile_url":"https://independent.academia.edu/GreathouseCharles?f_ri=302","photo":"https://0.academia-photos.com/53353087/14114825/15116986/s65_charles.greathouse.jpg"},{"id":53427258,"first_name":"Vladimir","last_name":"Shevelev","domain_name":"independent","page_name":"VShevelev","display_name":"Vladimir Shevelev","profile_url":"https://independent.academia.edu/VShevelev?f_ri=302","photo":"/images/s65_no_pic.png"}],"research_interests":[{"id":302,"name":"Analytic Number Theory","url":"https://www.academia.edu/Documents/in/Analytic_Number_Theory?f_ri=302","nofollow":true},{"id":72443,"name":"Prime Numbers","url":"https://www.academia.edu/Documents/in/Prime_Numbers?f_ri=302","nofollow":true}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_32348153 coauthored" data-work_id="32348153" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/32348153/The_Repeated_Divisor_Function_and_Possible_Correlation_with_Highly_Composite_Numbers">The Repeated Divisor Function and Possible Correlation with Highly Composite Numbers</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest">Let n be a non-null positive integer and d(n) is the number of positive divisors of n, called the divisor function. Of course, d(n) ≤ n. d(n) = 1 if and only if n = 1.</div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/32348153" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="2bfb0ede5658f79595ad96c589c16bda" rel="nofollow" data-download="{"attachment_id":52556799,"asset_id":32348153,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/52556799/download_file?st=MTc0MDg2MzQyNyw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="17189940" href="https://independent.academia.edu/sandipchakrabarty">Sayak Chakrabarty</a><script data-card-contents-for-user="17189940" type="text/json">{"id":17189940,"first_name":"Sayak","last_name":"Chakrabarty","domain_name":"independent","page_name":"sandipchakrabarty","display_name":"Sayak Chakrabarty","profile_url":"https://independent.academia.edu/sandipchakrabarty?f_ri=302","photo":"https://0.academia-photos.com/17189940/4745316/5471096/s65_sandip.chakrabarty.jpg"}</script></span></span><span class="u-displayInlineBlock InlineList-item-text"> and <span class="u-textDecorationUnderline u-clickable InlineList-item-text js-work-more-authors-32348153">+1</span><div class="hidden js-additional-users-32348153"><div><span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a href="https://independent.academia.edu/ArghyaDatta1">Arghya Datta</a></span></div></div></span><script>(function(){ var popoverSettings = { el: $('.js-work-more-authors-32348153'), placement: 'bottom', hide_delay: 200, html: true, content: function(){ return $('.js-additional-users-32348153').html(); } } new HoverPopover(popoverSettings); })();</script></li><li class="js-paper-rank-work_32348153 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="32348153"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 32348153, container: ".js-paper-rank-work_32348153", }); });</script></li><li class="js-percentile-work_32348153 InlineList-item InlineList-item--bordered hidden u-tcGrayDark"><span class="percentile-widget hidden"><span class="u-mr2x percentile-widget" style="display: none">•</span><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 32348153; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-percentile-work_32348153"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></li><li class="js-view-count-work_32348153 InlineList-item InlineList-item--bordered hidden"><div><span><span class="js-view-count view-count u-mr2x" data-work-id="32348153"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 32348153; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=32348153]").text(description); $(".js-view-count-work_32348153").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_32348153").removeClass('hidden') })</script></div></li><li class="InlineList-item u-positionRelative" style="max-width: 250px"><div class="u-positionAbsolute" data-has-card-for-ri-list="32348153"><i class="fa fa-tag InlineList-item-icon u-positionRelative"></i></div><span class="InlineList-item-text u-textTruncate u-pl6x"><a class="InlineList-item-text" data-has-card-for-ri="302" rel="nofollow" href="https://www.academia.edu/Documents/in/Analytic_Number_Theory">Analytic Number Theory</a><script data-card-contents-for-ri="302" type="text/json">{"id":302,"name":"Analytic Number Theory","url":"https://www.academia.edu/Documents/in/Analytic_Number_Theory?f_ri=302","nofollow":true}</script></span></li><script>(function(){ if (false) { new Aedu.ResearchInterestListCard({ el: $('*[data-has-card-for-ri-list=32348153]'), work: {"id":32348153,"title":"The Repeated Divisor Function and Possible Correlation with Highly Composite Numbers","created_at":"2017-04-09T04:16:26.695-07:00","url":"https://www.academia.edu/32348153/The_Repeated_Divisor_Function_and_Possible_Correlation_with_Highly_Composite_Numbers?f_ri=302","dom_id":"work_32348153","summary":"Let n be a non-null positive integer and d(n) is the number of positive divisors of n, called the divisor function. Of course, d(n) ≤ n. d(n) = 1 if and only if n = 1.","downloadable_attachments":[{"id":52556799,"asset_id":32348153,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":17189940,"first_name":"Sayak","last_name":"Chakrabarty","domain_name":"independent","page_name":"sandipchakrabarty","display_name":"Sayak Chakrabarty","profile_url":"https://independent.academia.edu/sandipchakrabarty?f_ri=302","photo":"https://0.academia-photos.com/17189940/4745316/5471096/s65_sandip.chakrabarty.jpg"},{"id":62960041,"first_name":"Arghya","last_name":"Datta","domain_name":"independent","page_name":"ArghyaDatta1","display_name":"Arghya Datta","profile_url":"https://independent.academia.edu/ArghyaDatta1?f_ri=302","photo":"https://0.academia-photos.com/62960041/17128645/17284456/s65_arghya.datta.jpg"}],"research_interests":[{"id":302,"name":"Analytic Number Theory","url":"https://www.academia.edu/Documents/in/Analytic_Number_Theory?f_ri=302","nofollow":true}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_36721110" data-work_id="36721110" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/36721110/The_Functional_Equation_for_Bernoulli_Numbers">The Functional Equation for Bernoulli Numbers</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest"><div class="summarized">Bernoulli numbers which are ubiquitous in mathematics, typically appearing either as the Taylor coefficients of x/ tan x or else being very closed to this, as special values of the Riemann zeta function. But they also sometimes appear in... <a class="more_link u-tcGrayDark u-linkUnstyled" data-container=".work_36721110" data-show=".complete" data-hide=".summarized" data-more-link-behavior="true" href="#">more</a></div><div class="complete hidden">Bernoulli numbers which are ubiquitous in mathematics, typically appearing either as the Taylor coefficients of x/ tan x or else being very closed to this, as special values of the Riemann zeta function. But they also sometimes appear in other guises and in other combinations. Here we will look at the functional equation satisfied by the Bernoulli numbers. We begin with the less familiar and divergent ordinary generating series β(x) = ∞ n=0</div></div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/36721110" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="90e1390b5fd14e5f75cc870a077506e7" rel="nofollow" data-download="{"attachment_id":56662273,"asset_id":36721110,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/56662273/download_file?st=MTc0MDg2MzQyNyw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="2792810" href="https://independentresearcher.academia.edu/DrJMAshfaqueAMIMAMInstP">Dr. J. 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But they also sometimes appear in other guises and in other combinations. Here we will look at the functional equation satisfied by the Bernoulli numbers. We begin with the less familiar and divergent ordinary generating series β(x) = ∞ n=0","downloadable_attachments":[{"id":56662273,"asset_id":36721110,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":2792810,"first_name":"Dr. J. M.","last_name":"Ashfaque (MInstP)","domain_name":"independentresearcher","page_name":"DrJMAshfaqueAMIMAMInstP","display_name":"Dr. J. M. 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system.","downloadable_attachments":[{"id":53612948,"asset_id":33591980,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":51072244,"first_name":"Craig","last_name":"Feinstein","domain_name":"independent","page_name":"CraigFeinstein","display_name":"Craig Feinstein","profile_url":"https://independent.academia.edu/CraigFeinstein?f_ri=302","photo":"/images/s65_no_pic.png"}],"research_interests":[{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=302","nofollow":true},{"id":302,"name":"Analytic Number Theory","url":"https://www.academia.edu/Documents/in/Analytic_Number_Theory?f_ri=302","nofollow":true}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_17411960" data-work_id="17411960" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 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Kaleff","profile_url":"https://uff.academia.edu/AnaKaleff?f_ri=302","photo":"https://0.academia-photos.com/3983664/3331925/3920603/s65_ana.kaleff.jpg"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics?f_ri=302","nofollow":true},{"id":302,"name":"Analytic Number Theory","url":"https://www.academia.edu/Documents/in/Analytic_Number_Theory?f_ri=302","nofollow":true},{"id":592394,"name":"Analisis","url":"https://www.academia.edu/Documents/in/Analisis?f_ri=302","nofollow":true}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_71423609" data-work_id="71423609" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/71423609/The_Significance_Of_The_Non_Trivial_Zeros_Of_The_Riemann_Zeta_Function">The Significance Of The Non-Trivial Zeros Of The Riemann Zeta Function</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest"><div class="summarized">This paper expounds the role of the non-trivial zeros of the Riemann zeta function ζ and supplements the author’s earlier papers on the Riemann hypothesis. There is a lot of mystery surrounding the non-trivial zeros. MSC: 11-XX (Number... <a class="more_link u-tcGrayDark u-linkUnstyled" data-container=".work_71423609" data-show=".complete" data-hide=".summarized" data-more-link-behavior="true" href="#">more</a></div><div class="complete hidden">This paper expounds the role of the non-trivial zeros of the Riemann zeta function ζ and supplements the author’s earlier papers on the Riemann hypothesis. There is a lot of mystery surrounding the non-trivial zeros. MSC: 11-XX (Number Theory)</div></div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/71423609" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="36689f37d04972be0fd169a467a38f42" rel="nofollow" data-download="{"attachment_id":80772912,"asset_id":71423609,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/80772912/download_file?st=MTc0MDg2MzQyNyw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="11278844" href="https://independent.academia.edu/BertrandWong">Bertrand Wong</a><script data-card-contents-for-user="11278844" type="text/json">{"id":11278844,"first_name":"Bertrand","last_name":"Wong","domain_name":"independent","page_name":"BertrandWong","display_name":"Bertrand Wong","profile_url":"https://independent.academia.edu/BertrandWong?f_ri=302","photo":"https://0.academia-photos.com/11278844/173241083/163252381/s65_bertrand.wong.png"}</script></span></span></li><li class="js-paper-rank-work_71423609 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="71423609"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 71423609, container: ".js-paper-rank-work_71423609", }); 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There is a lot of mystery surrounding the non-trivial zeros. MSC: 11-XX (Number Theory)","downloadable_attachments":[{"id":80772912,"asset_id":71423609,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":11278844,"first_name":"Bertrand","last_name":"Wong","domain_name":"independent","page_name":"BertrandWong","display_name":"Bertrand Wong","profile_url":"https://independent.academia.edu/BertrandWong?f_ri=302","photo":"https://0.academia-photos.com/11278844/173241083/163252381/s65_bertrand.wong.png"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics?f_ri=302","nofollow":true},{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=302","nofollow":true},{"id":302,"name":"Analytic Number Theory","url":"https://www.academia.edu/Documents/in/Analytic_Number_Theory?f_ri=302","nofollow":true},{"id":303,"name":"Algebraic Number Theory","url":"https://www.academia.edu/Documents/in/Algebraic_Number_Theory?f_ri=302","nofollow":true},{"id":371,"name":"Functional Analysis","url":"https://www.academia.edu/Documents/in/Functional_Analysis?f_ri=302"},{"id":19997,"name":"Pure Mathematics","url":"https://www.academia.edu/Documents/in/Pure_Mathematics?f_ri=302"},{"id":72443,"name":"Prime Numbers","url":"https://www.academia.edu/Documents/in/Prime_Numbers?f_ri=302"},{"id":153546,"name":"Primes","url":"https://www.academia.edu/Documents/in/Primes?f_ri=302"},{"id":325323,"name":"Functions","url":"https://www.academia.edu/Documents/in/Functions?f_ri=302"}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_30963764" data-work_id="30963764" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/30963764/The_Fundamental_Critique_of_Mathematics">The Fundamental Critique of Mathematics</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest">This paper aims to illuminate the fundamental flaws of the basic theory of arithmetic. No further introduction needed.</div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/30963764" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="7a33907109b9d689e828fcee6d0e8b2c" rel="nofollow" data-download="{"attachment_id":51396233,"asset_id":30963764,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/51396233/download_file?st=MTc0MDg2MzQyNyw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="5746293" href="https://southernct.academia.edu/NathanCoppedge">Nathan Coppedge</a><script data-card-contents-for-user="5746293" type="text/json">{"id":5746293,"first_name":"Nathan","last_name":"Coppedge","domain_name":"southernct","page_name":"NathanCoppedge","display_name":"Nathan Coppedge","profile_url":"https://southernct.academia.edu/NathanCoppedge?f_ri=302","photo":"https://0.academia-photos.com/5746293/2482759/76140274/s65_nathan.coppedge.jpg"}</script></span></span></li><li class="js-paper-rank-work_30963764 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="30963764"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 30963764, container: ".js-paper-rank-work_30963764", }); 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No further introduction needed.","downloadable_attachments":[{"id":51396233,"asset_id":30963764,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":5746293,"first_name":"Nathan","last_name":"Coppedge","domain_name":"southernct","page_name":"NathanCoppedge","display_name":"Nathan Coppedge","profile_url":"https://southernct.academia.edu/NathanCoppedge?f_ri=302","photo":"https://0.academia-photos.com/5746293/2482759/76140274/s65_nathan.coppedge.jpg"}],"research_interests":[{"id":8,"name":"Critical Theory","url":"https://www.academia.edu/Documents/in/Critical_Theory?f_ri=302","nofollow":true},{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics?f_ri=302","nofollow":true},{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=302","nofollow":true},{"id":302,"name":"Analytic Number Theory","url":"https://www.academia.edu/Documents/in/Analytic_Number_Theory?f_ri=302","nofollow":true},{"id":303,"name":"Algebraic Number Theory","url":"https://www.academia.edu/Documents/in/Algebraic_Number_Theory?f_ri=302"},{"id":329,"name":"Algebra","url":"https://www.academia.edu/Documents/in/Algebra?f_ri=302"},{"id":360,"name":"Logic And Foundations Of Mathematics","url":"https://www.academia.edu/Documents/in/Logic_And_Foundations_Of_Mathematics?f_ri=302"},{"id":363,"name":"Set Theory","url":"https://www.academia.edu/Documents/in/Set_Theory?f_ri=302"},{"id":803,"name":"Philosophy","url":"https://www.academia.edu/Documents/in/Philosophy?f_ri=302"},{"id":815,"name":"Epistemology","url":"https://www.academia.edu/Documents/in/Epistemology?f_ri=302"},{"id":2509,"name":"Philosophy Of Mathematics","url":"https://www.academia.edu/Documents/in/Philosophy_Of_Mathematics?f_ri=302"},{"id":10957,"name":"Conventions","url":"https://www.academia.edu/Documents/in/Conventions?f_ri=302"},{"id":13279,"name":"Philosophy of Logic","url":"https://www.academia.edu/Documents/in/Philosophy_of_Logic?f_ri=302"},{"id":20254,"name":"Elementary Number Theory","url":"https://www.academia.edu/Documents/in/Elementary_Number_Theory?f_ri=302"},{"id":29972,"name":"Linear Algebra","url":"https://www.academia.edu/Documents/in/Linear_Algebra?f_ri=302"},{"id":39009,"name":"Set Theory (Philosophy)","url":"https://www.academia.edu/Documents/in/Set_Theory_Philosophy_?f_ri=302"},{"id":39056,"name":"Theoretical Mathematics","url":"https://www.academia.edu/Documents/in/Theoretical_Mathematics?f_ri=302"},{"id":40241,"name":"Numeracy","url":"https://www.academia.edu/Documents/in/Numeracy?f_ri=302"},{"id":44444,"name":"Foundationalism","url":"https://www.academia.edu/Documents/in/Foundationalism?f_ri=302"},{"id":68570,"name":"Formalism","url":"https://www.academia.edu/Documents/in/Formalism?f_ri=302"},{"id":118005,"name":"Axiomatic Set Theory","url":"https://www.academia.edu/Documents/in/Axiomatic_Set_Theory?f_ri=302"},{"id":131903,"name":"Arithmetic","url":"https://www.academia.edu/Documents/in/Arithmetic?f_ri=302"},{"id":142963,"name":"Eccentricity","url":"https://www.academia.edu/Documents/in/Eccentricity?f_ri=302"},{"id":157512,"name":"Arithmetics","url":"https://www.academia.edu/Documents/in/Arithmetics?f_ri=302"},{"id":310605,"name":"Epistemological Foundationalism","url":"https://www.academia.edu/Documents/in/Epistemological_Foundationalism?f_ri=302"},{"id":402213,"name":"Basic Mathematics","url":"https://www.academia.edu/Documents/in/Basic_Mathematics?f_ri=302"},{"id":740756,"name":"Number System","url":"https://www.academia.edu/Documents/in/Number_System?f_ri=302"},{"id":781100,"name":"Logic and Foundations of Mathematics; 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Its secret is the secret of a forgotten body, the seventh in the series of multifaceted as Life, Seven. We know six of it now. But the... <a class="more_link u-tcGrayDark u-linkUnstyled" data-container=".work_7389763" data-show=".complete" data-hide=".summarized" data-more-link-behavior="true" href="#">more</a></div><div class="complete hidden">People these days know the Universe as a Whole, because not knowing the edge between This and That. Its secret is the secret of a forgotten body, the seventh in the series of multifaceted as Life, Seven. We know six of it now. But the Ancestors knew all seven. // Люди ныне не знают Вселенной как Целого, не зная грани меж Этим и Тем. Тайна  ее есть тайна забытого тела, седьмого в строю многогранных как Жизнь, Семь. Мы знаем шесть ныне. 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href="https://www.academia.edu/4088859/Elementary_Arithmetic_in_Base_7">Elementary Arithmetic in Base 7</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest"><div class="summarized">"Elementary Arithmetic in Base 7" presents a novel framework that integrates numerical systems with color representation through a unique chromatic approach based on cyclic orders. The concept revolves around the relationship between... <a class="more_link u-tcGrayDark u-linkUnstyled" data-container=".work_4088859" data-show=".complete" data-hide=".summarized" data-more-link-behavior="true" href="#">more</a></div><div class="complete hidden">"Elementary Arithmetic in Base 7" presents a novel framework that integrates numerical systems with color representation through a unique chromatic approach based on cyclic orders. The concept revolves around the relationship between different numerical bases and their visual representations, specifically focusing on Base 7. The paper explores the mathematical properties of cyclic numbers, monotonic functions, and their implications for establishing a harmonic system that maps numbers to colors in a structured way, thereby creating a cohesive language for sounds, colors, and numbers. For more information see the following link: <a href="https://pdpseven.wixsite.com/sound-color" rel="nofollow">https://pdpseven.wixsite.com/sound-color</a>. <br />For more information see the following link: <br /><a href="https://pdpseven.wixsite.com/sound-color" rel="nofollow">https://pdpseven.wixsite.com/sound-color</a></div></div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/4088859" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="1930006" href="https://independent.academia.edu/paolodipasquale">Paolo Di Pasquale</a><script data-card-contents-for-user="1930006" type="text/json">{"id":1930006,"first_name":"Paolo","last_name":"Di Pasquale","domain_name":"independent","page_name":"paolodipasquale","display_name":"Paolo Di Pasquale","profile_url":"https://independent.academia.edu/paolodipasquale?f_ri=302","photo":"https://0.academia-photos.com/1930006/651321/807219/s65_paolo.di_pasquale.jpg"}</script></span></span></li><li class="js-paper-rank-work_4088859 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="4088859"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 4088859, container: ".js-paper-rank-work_4088859", }); 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itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/3221297/On_Plouffes_Ramanujan_Identities">On Plouffe's Ramanujan Identities</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest">Recently, Simon Plouffe has discovered a number of identities for the Riemann zeta function at odd integer values. These identities are obtained numerically and are inspired by a prototypical series for Apéry's constant given by Ramanujan:</div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/3221297" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="7d93a81441b1f65d29ac4e6d94d6142d" rel="nofollow" data-download="{"attachment_id":31090809,"asset_id":3221297,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/31090809/download_file?st=MTc0MDg2MzQyOCw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="3685531" href="https://independent.academia.edu/LinasVepstas">Linas Vepstas</a><script data-card-contents-for-user="3685531" type="text/json">{"id":3685531,"first_name":"Linas","last_name":"Vepstas","domain_name":"independent","page_name":"LinasVepstas","display_name":"Linas Vepstas","profile_url":"https://independent.academia.edu/LinasVepstas?f_ri=302","photo":"https://gravatar.com/avatar/a221b4b4e6dd199750731c51ee864a91?s=65"}</script></span></span></li><li class="js-paper-rank-work_3221297 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="3221297"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 3221297, container: ".js-paper-rank-work_3221297", }); });</script></li><li class="js-percentile-work_3221297 InlineList-item InlineList-item--bordered hidden u-tcGrayDark"><span class="percentile-widget hidden"><span class="u-mr2x percentile-widget" style="display: none">•</span><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 3221297; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-percentile-work_3221297"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></li><li class="js-view-count-work_3221297 InlineList-item InlineList-item--bordered hidden"><div><span><span class="js-view-count view-count u-mr2x" data-work-id="3221297"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 3221297; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=3221297]").text(description); $(".js-view-count-work_3221297").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_3221297").removeClass('hidden') })</script></div></li><li class="InlineList-item u-positionRelative" style="max-width: 250px"><div class="u-positionAbsolute" data-has-card-for-ri-list="3221297"><i class="fa fa-tag InlineList-item-icon u-positionRelative"></i>  <a class="InlineList-item-text u-positionRelative">2</a>  </div><span class="InlineList-item-text u-textTruncate u-pl9x"><a class="InlineList-item-text" data-has-card-for-ri="302" rel="nofollow" href="https://www.academia.edu/Documents/in/Analytic_Number_Theory">Analytic Number Theory</a>, <script data-card-contents-for-ri="302" type="text/json">{"id":302,"name":"Analytic Number Theory","url":"https://www.academia.edu/Documents/in/Analytic_Number_Theory?f_ri=302","nofollow":true}</script><a class="InlineList-item-text" data-has-card-for-ri="51368" rel="nofollow" href="https://www.academia.edu/Documents/in/Srinivasa_Ramanujan">Srinivasa Ramanujan</a><script data-card-contents-for-ri="51368" type="text/json">{"id":51368,"name":"Srinivasa Ramanujan","url":"https://www.academia.edu/Documents/in/Srinivasa_Ramanujan?f_ri=302","nofollow":true}</script></span></li><script>(function(){ if (true) { new Aedu.ResearchInterestListCard({ el: $('*[data-has-card-for-ri-list=3221297]'), work: {"id":3221297,"title":"On Plouffe's Ramanujan Identities","created_at":"2013-04-05T14:05:56.457-07:00","url":"https://www.academia.edu/3221297/On_Plouffes_Ramanujan_Identities?f_ri=302","dom_id":"work_3221297","summary":"Recently, Simon Plouffe has discovered a number of identities for the Riemann zeta function at odd integer values. These identities are obtained numerically and are inspired by a prototypical series for Apéry's constant given by Ramanujan:","downloadable_attachments":[{"id":31090809,"asset_id":3221297,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":3685531,"first_name":"Linas","last_name":"Vepstas","domain_name":"independent","page_name":"LinasVepstas","display_name":"Linas Vepstas","profile_url":"https://independent.academia.edu/LinasVepstas?f_ri=302","photo":"https://gravatar.com/avatar/a221b4b4e6dd199750731c51ee864a91?s=65"}],"research_interests":[{"id":302,"name":"Analytic Number Theory","url":"https://www.academia.edu/Documents/in/Analytic_Number_Theory?f_ri=302","nofollow":true},{"id":51368,"name":"Srinivasa Ramanujan","url":"https://www.academia.edu/Documents/in/Srinivasa_Ramanujan?f_ri=302","nofollow":true}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_46878315" data-work_id="46878315" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/46878315/Number_Theory">Number Theory</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest"><div class="summarized">Number theory is a branch of mathematics that is primarily focused on the study of positive integers, or natural numbers, and their properties such as divisibility, prime factorization, or solvability of equations in integers. Number... <a class="more_link u-tcGrayDark u-linkUnstyled" data-container=".work_46878315" data-show=".complete" data-hide=".summarized" data-more-link-behavior="true" href="#">more</a></div><div class="complete hidden">Number theory is a branch of mathematics that is primarily focused on the study of positive integers, or natural numbers, and their properties such as divisibility, prime factorization, or solvability of equations in integers. Number theory has a very long and diverse history, and some of the greatest mathematicians of all time, such as Euclid, Euler and Gauss, have made significant contributions to it. Throughout its long history, number theory has often been considered as the "purest" branch of mathematics in the sense that it was the furthest from any concrete application. However, a significant change took place in the mid-1970s, and nowadays, number theory is one of the most important branches of mathematics for applications in cryptography and secure information exchange. This book is based on teaching materials from the courses Number Theory and Elementary Number Theory, which are taught at the undergraduate level studies at the Department of Mathematics, Faculty of Science, University of Zagreb, and the courses Diophantine Equations and Diophantine Approximations and Applications, which were taught at the doctoral program of mathematics at that faculty. The book thoroughly covers the content of these courses, but it also contains other related topics such as elliptic curves, which are the subject of the last two chapters in the book. The book also provides an insight into subjects that were and are at the centre of research interest of the author of the book and other members of the Croatian group in number theory, gathered around the Seminar on Number Theory and Algebra. This book is primarily intended for students of mathematics and related faculties who attend courses in number theory and its applications. However, it can also be useful to advanced high school students who are preparing for mathematics competitions in which at all levels, from the school level to international competitions, number theory has a significant role, and for doctoral students and scientists in the fields of number theory, algebra and cryptography. In the English edition, there are only minor changes compared with the Croatian version. Several misprints noticed by the author and the readers were corrected. Some information and references were updated, in particular, those related to elliptic curves rank records and new constructions of families of rational Diophantine sextuples. At just a few places in the Croatian version of the book only the references to literature in Croatian were given ; these references were expanded in the English edition with the appropriate recommendations of literature in English. The list of references has been expanded to include some recent books and papers, as well as some references which were mentioned in the text of the Croatian edition but were not included in the list of references.</div></div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/46878315" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="f2a3ee20f36c2ca433f9495dee932e78" rel="nofollow" data-download="{"attachment_id":66270348,"asset_id":46878315,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/66270348/download_file?st=MTc0MDg2MzQyOCw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="29857" href="https://unizg.academia.edu/AndrejDujella">Andrej Dujella</a><script data-card-contents-for-user="29857" type="text/json">{"id":29857,"first_name":"Andrej","last_name":"Dujella","domain_name":"unizg","page_name":"AndrejDujella","display_name":"Andrej Dujella","profile_url":"https://unizg.academia.edu/AndrejDujella?f_ri=302","photo":"https://0.academia-photos.com/29857/9709/9189/s65_andrej.dujella.jpg"}</script></span></span></li><li class="js-paper-rank-work_46878315 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="46878315"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 46878315, container: ".js-paper-rank-work_46878315", }); 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$(".js-view-count[data-work-id=46878315]").text(description); $(".js-view-count-work_46878315").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_46878315").removeClass('hidden') })</script></div></li><li class="InlineList-item u-positionRelative" style="max-width: 250px"><div class="u-positionAbsolute" data-has-card-for-ri-list="46878315"><i class="fa fa-tag InlineList-item-icon u-positionRelative"></i>  <a class="InlineList-item-text u-positionRelative">8</a>  </div><span class="InlineList-item-text u-textTruncate u-pl9x"><a class="InlineList-item-text" data-has-card-for-ri="301" rel="nofollow" href="https://www.academia.edu/Documents/in/Number_Theory">Number Theory</a>, <script data-card-contents-for-ri="301" type="text/json">{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=302","nofollow":true}</script><a class="InlineList-item-text" data-has-card-for-ri="302" rel="nofollow" href="https://www.academia.edu/Documents/in/Analytic_Number_Theory">Analytic Number Theory</a>, <script data-card-contents-for-ri="302" type="text/json">{"id":302,"name":"Analytic Number Theory","url":"https://www.academia.edu/Documents/in/Analytic_Number_Theory?f_ri=302","nofollow":true}</script><a class="InlineList-item-text" data-has-card-for-ri="303" rel="nofollow" href="https://www.academia.edu/Documents/in/Algebraic_Number_Theory">Algebraic Number Theory</a>, <script data-card-contents-for-ri="303" type="text/json">{"id":303,"name":"Algebraic Number Theory","url":"https://www.academia.edu/Documents/in/Algebraic_Number_Theory?f_ri=302","nofollow":true}</script><a class="InlineList-item-text" data-has-card-for-ri="16542" rel="nofollow" href="https://www.academia.edu/Documents/in/Cryptography">Cryptography</a><script data-card-contents-for-ri="16542" type="text/json">{"id":16542,"name":"Cryptography","url":"https://www.academia.edu/Documents/in/Cryptography?f_ri=302","nofollow":true}</script></span></li><script>(function(){ if (true) { new Aedu.ResearchInterestListCard({ el: $('*[data-has-card-for-ri-list=46878315]'), work: {"id":46878315,"title":"Number Theory","created_at":"2021-04-14T21:37:18.008-07:00","url":"https://www.academia.edu/46878315/Number_Theory?f_ri=302","dom_id":"work_46878315","summary":"Number theory is a branch of mathematics that is primarily focused on the study of positive integers, or natural numbers, and their properties such as divisibility, prime factorization, or solvability of equations in integers. Number theory has a very long and diverse history, and some of the greatest mathematicians of all time, such as Euclid, Euler and Gauss, have made significant contributions to it. Throughout its long history, number theory has often been considered as the \"purest\" branch of mathematics in the sense that it was the furthest from any concrete application. However, a significant change took place in the mid-1970s, and nowadays, number theory is one of the most important branches of mathematics for applications in cryptography and secure information exchange. This book is based on teaching materials from the courses Number Theory and Elementary Number Theory, which are taught at the undergraduate level studies at the Department of Mathematics, Faculty of Science, University of Zagreb, and the courses Diophantine Equations and Diophantine Approximations and Applications, which were taught at the doctoral program of mathematics at that faculty. The book thoroughly covers the content of these courses, but it also contains other related topics such as elliptic curves, which are the subject of the last two chapters in the book. The book also provides an insight into subjects that were and are at the centre of research interest of the author of the book and other members of the Croatian group in number theory, gathered around the Seminar on Number Theory and Algebra. This book is primarily intended for students of mathematics and related faculties who attend courses in number theory and its applications. However, it can also be useful to advanced high school students who are preparing for mathematics competitions in which at all levels, from the school level to international competitions, number theory has a significant role, and for doctoral students and scientists in the fields of number theory, algebra and cryptography. In the English edition, there are only minor changes compared with the Croatian version. Several misprints noticed by the author and the readers were corrected. Some information and references were updated, in particular, those related to elliptic curves rank records and new constructions of families of rational Diophantine sextuples. At just a few places in the Croatian version of the book only the references to literature in Croatian were given ; these references were expanded in the English edition with the appropriate recommendations of literature in English. The list of references has been expanded to include some recent books and papers, as well as some references which were mentioned in the text of the Croatian edition but were not included in the list of references.","downloadable_attachments":[{"id":66270348,"asset_id":46878315,"asset_type":"Work","always_allow_download":false},{"id":66270306,"asset_id":46878315,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":29857,"first_name":"Andrej","last_name":"Dujella","domain_name":"unizg","page_name":"AndrejDujella","display_name":"Andrej Dujella","profile_url":"https://unizg.academia.edu/AndrejDujella?f_ri=302","photo":"https://0.academia-photos.com/29857/9709/9189/s65_andrej.dujella.jpg"}],"research_interests":[{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=302","nofollow":true},{"id":302,"name":"Analytic Number Theory","url":"https://www.academia.edu/Documents/in/Analytic_Number_Theory?f_ri=302","nofollow":true},{"id":303,"name":"Algebraic Number Theory","url":"https://www.academia.edu/Documents/in/Algebraic_Number_Theory?f_ri=302","nofollow":true},{"id":16542,"name":"Cryptography","url":"https://www.academia.edu/Documents/in/Cryptography?f_ri=302","nofollow":true},{"id":42213,"name":"Diophantine approximation","url":"https://www.academia.edu/Documents/in/Diophantine_approximation?f_ri=302"},{"id":44397,"name":"Elliptic curves","url":"https://www.academia.edu/Documents/in/Elliptic_curves?f_ri=302"},{"id":969030,"name":"Polynomials","url":"https://www.academia.edu/Documents/in/Polynomials?f_ri=302"},{"id":1438521,"name":"Diophantine Equations","url":"https://www.academia.edu/Documents/in/Diophantine_Equations?f_ri=302"}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_69939690" data-work_id="69939690" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/69939690/A_New_Family_of_the_%CE%BB_Generalized_Hurwitz_Lerch_Zeta_Functions_with_Applications">A New Family of the λ -Generalized Hurwitz-Lerch Zeta Functions with Applications</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest"><div class="summarized">Motivated largely by a number of recent investigations, we introduce and investigate the various properties of a certain new family of the λ-generalized Hurwitz-Lerch zeta functions. We derive many potentially useful results involving... <a class="more_link u-tcGrayDark u-linkUnstyled" data-container=".work_69939690" data-show=".complete" data-hide=".summarized" data-more-link-behavior="true" href="#">more</a></div><div class="complete hidden">Motivated largely by a number of recent investigations, we introduce and investigate the various properties of a certain new family of the λ-generalized Hurwitz-Lerch zeta functions. We derive many potentially useful results involving these λ-generalized Hurwitz-Lerch zeta functions including (for example) their partial differential equations, new series and Mellin-Barnes type contour integral representations (which are associated with Fox's H-function) and several other summation formulas. We discuss their potential application in Number Theory by appropriately constructing a seemingly novel continuous analogue of Lippert's Hurwitz measure. We also consider some other statistical applications of the family of the λ-generalized Hurwitz-Lerch zeta functions in probability distribution theory.</div></div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/69939690" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="9fd9b3063d5249ef857fa18dcdb7ba3c" rel="nofollow" data-download="{"attachment_id":79846349,"asset_id":69939690,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/79846349/download_file?st=MTc0MDg2MzQyOCw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="38145660" href="https://uvic.academia.edu/HMSrivastava">H. M. Srivastava</a><script data-card-contents-for-user="38145660" type="text/json">{"id":38145660,"first_name":"H. M.","last_name":"Srivastava","domain_name":"uvic","page_name":"HMSrivastava","display_name":"H. M. 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We derive many potentially useful results involving these λ-generalized Hurwitz-Lerch zeta functions including (for example) their partial differential equations, new series and Mellin-Barnes type contour integral representations (which are associated with Fox's H-function) and several other summation formulas. We discuss their potential application in Number Theory by appropriately constructing a seemingly novel continuous analogue of Lippert's Hurwitz measure. We also consider some other statistical applications of the family of the λ-generalized Hurwitz-Lerch zeta functions in probability distribution theory.","downloadable_attachments":[{"id":79846349,"asset_id":69939690,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":38145660,"first_name":"H. M.","last_name":"Srivastava","domain_name":"uvic","page_name":"HMSrivastava","display_name":"H. M. Srivastava","profile_url":"https://uvic.academia.edu/HMSrivastava?f_ri=302","photo":"https://0.academia-photos.com/38145660/10667201/11908456/s65_h._m..srivastava.jpg"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics?f_ri=302","nofollow":true},{"id":302,"name":"Analytic Number Theory","url":"https://www.academia.edu/Documents/in/Analytic_Number_Theory?f_ri=302","nofollow":true},{"id":305,"name":"Applied Mathematics","url":"https://www.academia.edu/Documents/in/Applied_Mathematics?f_ri=302","nofollow":true},{"id":377,"name":"Partial Differential Equations","url":"https://www.academia.edu/Documents/in/Partial_Differential_Equations?f_ri=302","nofollow":true},{"id":3370086,"name":"Probability Density Functions","url":"https://www.academia.edu/Documents/in/Probability_Density_Functions?f_ri=302"}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_5027791" data-work_id="5027791" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/5027791/Dirichlets_Theorem_and_Applications">Dirichlet's Theorem and Applications</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest"><div class="summarized">Dirichlet's theorem states that there exist an infinite number of primes in an arithmetic progression a + mk when a and m are relatively prime and k runs over the positive integers. While a few special cases of Dirichlet's theorem, such... <a class="more_link u-tcGrayDark u-linkUnstyled" data-container=".work_5027791" data-show=".complete" data-hide=".summarized" data-more-link-behavior="true" href="#">more</a></div><div class="complete hidden">Dirichlet's theorem states that there exist an infinite number of primes in an arithmetic progression a + mk when a and m are relatively prime and k runs over the positive integers. While a few special cases of Dirichlet's theorem, such as the arithmetic progression 2 + 3k, can be settled by elementary methods, the proof of the general case is much more involved. Analysis of the Riemann zeta-function and Dirichlet L-functions is used.</div></div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/5027791" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="8f3def1ebe3f4af0f64e4da571db529f" rel="nofollow" data-download="{"attachment_id":32260515,"asset_id":5027791,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/32260515/download_file?st=MTc0MDg2MzQyOCw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="6686566" href="https://indiana.academia.edu/NicholasStanford">Nicholas Stanford</a><script data-card-contents-for-user="6686566" type="text/json">{"id":6686566,"first_name":"Nicholas","last_name":"Stanford","domain_name":"indiana","page_name":"NicholasStanford","display_name":"Nicholas Stanford","profile_url":"https://indiana.academia.edu/NicholasStanford?f_ri=302","photo":"https://0.academia-photos.com/6686566/3022957/4423595/s65_nicholas.stanford.jpg"}</script></span></span></li><li class="js-paper-rank-work_5027791 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="5027791"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 5027791, container: ".js-paper-rank-work_5027791", }); 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While a few special cases of Dirichlet's theorem, such as the arithmetic progression 2 + 3k, can be settled by elementary methods, the proof of the general case is much more involved. Analysis of the Riemann zeta-function and Dirichlet L-functions is used.","downloadable_attachments":[{"id":32260515,"asset_id":5027791,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":6686566,"first_name":"Nicholas","last_name":"Stanford","domain_name":"indiana","page_name":"NicholasStanford","display_name":"Nicholas Stanford","profile_url":"https://indiana.academia.edu/NicholasStanford?f_ri=302","photo":"https://0.academia-photos.com/6686566/3022957/4423595/s65_nicholas.stanford.jpg"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics?f_ri=302","nofollow":true},{"id":302,"name":"Analytic Number Theory","url":"https://www.academia.edu/Documents/in/Analytic_Number_Theory?f_ri=302","nofollow":true}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_26212278" data-work_id="26212278" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/26212278/%CE%A3%CE%95%CE%99%CE%A1%CE%95%CE%A3_DIRICHLET_%CE%95%CE%99%CE%94%CE%99%CE%9A%CE%9F_%CE%98%CE%95%CE%9C%CE%91_pdf">ΣΕΙΡΕΣ DIRICHLET ΕΙΔΙΚΟ ΘΕΜΑ.pdf</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest"><div class="summarized">Ειδικό θέμα στις σειρές Dirichlet και στο Θεώρημα των Πρώτων αριθμών. Μελετούμε τις βασικές έννοιες των σειρών Dirichlet παρουσιάζοντας τα κύρια θεωρήματα , δίνουμε μία παραλλαγή της φόρμουλας του Perron, και κλείνουμε με την απόδειξη του... <a class="more_link u-tcGrayDark u-linkUnstyled" data-container=".work_26212278" data-show=".complete" data-hide=".summarized" data-more-link-behavior="true" href="#">more</a></div><div class="complete hidden">Ειδικό θέμα στις σειρές Dirichlet και στο Θεώρημα των Πρώτων αριθμών. Μελετούμε τις βασικές έννοιες των σειρών Dirichlet παρουσιάζοντας τα κύρια θεωρήματα , δίνουμε μία παραλλαγή της φόρμουλας του Perron, και κλείνουμε με την απόδειξη του Θεωρήματος των Πρώτων Αριθμών . Παραθέτουμε άλυτα προβλήματα στο τέλος της εργασίας.</div></div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/26212278" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="a3f76c9a9ac2b2ad689197869af2ab27" rel="nofollow" data-download="{"attachment_id":46532552,"asset_id":26212278,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/46532552/download_file?st=MTc0MDg2MzQyOCw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="11533266" href="https://auth.academia.edu/AthanasiosBeslikas">Athanasios Beslikas</a><script data-card-contents-for-user="11533266" type="text/json">{"id":11533266,"first_name":"Athanasios","last_name":"Beslikas","domain_name":"auth","page_name":"AthanasiosBeslikas","display_name":"Athanasios Beslikas","profile_url":"https://auth.academia.edu/AthanasiosBeslikas?f_ri=302","photo":"https://0.academia-photos.com/11533266/3348477/3940041/s65_athanasios.beslikas.jpg"}</script></span></span></li><li class="js-paper-rank-work_26212278 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="26212278"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 26212278, container: ".js-paper-rank-work_26212278", }); 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$(".js-view-count[data-work-id=26212278]").text(description); $(".js-view-count-work_26212278").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_26212278").removeClass('hidden') })</script></div></li><li class="InlineList-item u-positionRelative" style="max-width: 250px"><div class="u-positionAbsolute" data-has-card-for-ri-list="26212278"><i class="fa fa-tag InlineList-item-icon u-positionRelative"></i>  <a class="InlineList-item-text u-positionRelative">3</a>  </div><span class="InlineList-item-text u-textTruncate u-pl9x"><a class="InlineList-item-text" data-has-card-for-ri="300" rel="nofollow" href="https://www.academia.edu/Documents/in/Mathematics">Mathematics</a>, <script data-card-contents-for-ri="300" type="text/json">{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics?f_ri=302","nofollow":true}</script><a class="InlineList-item-text" data-has-card-for-ri="302" rel="nofollow" href="https://www.academia.edu/Documents/in/Analytic_Number_Theory">Analytic Number Theory</a>, <script data-card-contents-for-ri="302" type="text/json">{"id":302,"name":"Analytic Number Theory","url":"https://www.academia.edu/Documents/in/Analytic_Number_Theory?f_ri=302","nofollow":true}</script><a class="InlineList-item-text" data-has-card-for-ri="370" rel="nofollow" href="https://www.academia.edu/Documents/in/Complex_Analysis">Complex Analysis</a><script data-card-contents-for-ri="370" type="text/json">{"id":370,"name":"Complex Analysis","url":"https://www.academia.edu/Documents/in/Complex_Analysis?f_ri=302","nofollow":true}</script></span></li><script>(function(){ if (true) { new Aedu.ResearchInterestListCard({ el: $('*[data-has-card-for-ri-list=26212278]'), work: {"id":26212278,"title":"ΣΕΙΡΕΣ DIRICHLET ΕΙΔΙΚΟ ΘΕΜΑ.pdf","created_at":"2016-06-16T02:36:09.245-07:00","url":"https://www.academia.edu/26212278/%CE%A3%CE%95%CE%99%CE%A1%CE%95%CE%A3_DIRICHLET_%CE%95%CE%99%CE%94%CE%99%CE%9A%CE%9F_%CE%98%CE%95%CE%9C%CE%91_pdf?f_ri=302","dom_id":"work_26212278","summary":"Ειδικό θέμα στις σειρές Dirichlet και στο Θεώρημα των Πρώτων αριθμών. Μελετούμε τις βασικές έννοιες των σειρών Dirichlet παρουσιάζοντας τα κύρια θεωρήματα , δίνουμε μία παραλλαγή της φόρμουλας του Perron, και κλείνουμε με την απόδειξη του Θεωρήματος των Πρώτων Αριθμών . Παραθέτουμε άλυτα προβλήματα στο τέλος της εργασίας.","downloadable_attachments":[{"id":46532552,"asset_id":26212278,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":11533266,"first_name":"Athanasios","last_name":"Beslikas","domain_name":"auth","page_name":"AthanasiosBeslikas","display_name":"Athanasios Beslikas","profile_url":"https://auth.academia.edu/AthanasiosBeslikas?f_ri=302","photo":"https://0.academia-photos.com/11533266/3348477/3940041/s65_athanasios.beslikas.jpg"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics?f_ri=302","nofollow":true},{"id":302,"name":"Analytic Number Theory","url":"https://www.academia.edu/Documents/in/Analytic_Number_Theory?f_ri=302","nofollow":true},{"id":370,"name":"Complex Analysis","url":"https://www.academia.edu/Documents/in/Complex_Analysis?f_ri=302","nofollow":true}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_29580006" data-work_id="29580006" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/29580006/Numrat">Numrat</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest">all</div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/29580006" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="ab916adbaee9bb0e70251ec35b7f6b25" rel="nofollow" data-download="{"attachment_id":50024025,"asset_id":29580006,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/50024025/download_file?st=MTc0MDg2MzQyOCw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="55934902" href="https://independent.academia.edu/Sokolipamposhtur">Sokol i pamposhtur</a><script data-card-contents-for-user="55934902" type="text/json">{"id":55934902,"first_name":"Sokol","last_name":"i pamposhtur","domain_name":"independent","page_name":"Sokolipamposhtur","display_name":"Sokol i pamposhtur","profile_url":"https://independent.academia.edu/Sokolipamposhtur?f_ri=302","photo":"/images/s65_no_pic.png"}</script></span></span></li><li class="js-paper-rank-work_29580006 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="29580006"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 29580006, container: ".js-paper-rank-work_29580006", }); 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$(".js-view-count[data-work-id=29580006]").text(description); $(".js-view-count-work_29580006").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_29580006").removeClass('hidden') })</script></div></li><li class="InlineList-item u-positionRelative" style="max-width: 250px"><div class="u-positionAbsolute" data-has-card-for-ri-list="29580006"><i class="fa fa-tag InlineList-item-icon u-positionRelative"></i></div><span class="InlineList-item-text u-textTruncate u-pl6x"><a class="InlineList-item-text" data-has-card-for-ri="302" rel="nofollow" href="https://www.academia.edu/Documents/in/Analytic_Number_Theory">Analytic Number Theory</a><script data-card-contents-for-ri="302" type="text/json">{"id":302,"name":"Analytic Number Theory","url":"https://www.academia.edu/Documents/in/Analytic_Number_Theory?f_ri=302","nofollow":true}</script></span></li><script>(function(){ if (false) { new Aedu.ResearchInterestListCard({ el: $('*[data-has-card-for-ri-list=29580006]'), work: {"id":29580006,"title":"Numrat","created_at":"2016-11-01T00:19:14.331-07:00","url":"https://www.academia.edu/29580006/Numrat?f_ri=302","dom_id":"work_29580006","summary":"all","downloadable_attachments":[{"id":50024025,"asset_id":29580006,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":55934902,"first_name":"Sokol","last_name":"i pamposhtur","domain_name":"independent","page_name":"Sokolipamposhtur","display_name":"Sokol i pamposhtur","profile_url":"https://independent.academia.edu/Sokolipamposhtur?f_ri=302","photo":"/images/s65_no_pic.png"}],"research_interests":[{"id":302,"name":"Analytic Number Theory","url":"https://www.academia.edu/Documents/in/Analytic_Number_Theory?f_ri=302","nofollow":true}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_43855237" data-work_id="43855237" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/43855237/The_Closed_Form_of_Aperys_Constant">The Closed Form of Apery's Constant</a></div></div><div class="u-pb4x u-mt3x"></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/43855237" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="b1e7d8f204e42d1ae2b5fff73be4763d" rel="nofollow" data-download="{"attachment_id":64178541,"asset_id":43855237,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/64178541/download_file?st=MTc0MDg2MzQyOCw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="2792810" href="https://independentresearcher.academia.edu/DrJMAshfaqueAMIMAMInstP">Dr. J. M. Ashfaque (MInstP)</a><script data-card-contents-for-user="2792810" type="text/json">{"id":2792810,"first_name":"Dr. J. M.","last_name":"Ashfaque (MInstP)","domain_name":"independentresearcher","page_name":"DrJMAshfaqueAMIMAMInstP","display_name":"Dr. J. M. 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M.","last_name":"Ashfaque (MInstP)","domain_name":"independentresearcher","page_name":"DrJMAshfaqueAMIMAMInstP","display_name":"Dr. J. M. Ashfaque (MInstP)","profile_url":"https://independentresearcher.academia.edu/DrJMAshfaqueAMIMAMInstP?f_ri=302","photo":"https://0.academia-photos.com/2792810/914293/18370870/s65_dr._j._m..ashfaque_amima_minstp_.jpg"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics?f_ri=302","nofollow":true},{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=302","nofollow":true},{"id":302,"name":"Analytic Number Theory","url":"https://www.academia.edu/Documents/in/Analytic_Number_Theory?f_ri=302","nofollow":true},{"id":305,"name":"Applied Mathematics","url":"https://www.academia.edu/Documents/in/Applied_Mathematics?f_ri=302","nofollow":true},{"id":19997,"name":"Pure Mathematics","url":"https://www.academia.edu/Documents/in/Pure_Mathematics?f_ri=302"},{"id":20254,"name":"Elementary Number Theory","url":"https://www.academia.edu/Documents/in/Elementary_Number_Theory?f_ri=302"}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_38300048" data-work_id="38300048" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/38300048/Distribution_of_primes_the_computation_of_the_non_trivial_zeros_of_the_Riemann_zeta_function_root_identities_replication_identity_zero_free_zone_Weil_conjecture_and_new_mathematical_frontier">Distribution of primes, the computation of the non trivial zeros of the Riemann zeta function, root identities, replication identity, zero free zone, Weil conjecture and new mathematical frontier.</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest"><div class="summarized">In this paper we expand the prime number theorem, twice. Then we use both expansions to describe the distribution of primes. Afterwards we analyze the symmetry in the asymptotic equivalence. Before we come to the Weil conjecture, we... <a class="more_link u-tcGrayDark u-linkUnstyled" data-container=".work_38300048" data-show=".complete" data-hide=".summarized" data-more-link-behavior="true" href="#">more</a></div><div class="complete hidden">In this paper we expand the prime number theorem, twice. Then we use both expansions to describe the distribution of primes. Afterwards we analyze the symmetry in the asymptotic equivalence. Before we come to the Weil conjecture, we derive a solution of the non-trivial zeros of the Riemann zeta function, the root identity, which we expand to the entire complex plane via a replication identity and which we employ to show, that the critical strip next to the critical line is a zero free zone.</div></div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/38300048" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="bb67888c091d5e3b61421bdf88a4e852" rel="nofollow" data-download="{"attachment_id":87783785,"asset_id":38300048,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/87783785/download_file?st=MTc0MDg2MzQyOCw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="58174190" href="https://independent.academia.edu/MirceaEinstein">Mircea Einstein</a><script data-card-contents-for-user="58174190" type="text/json">{"id":58174190,"first_name":"Mircea","last_name":"Einstein","domain_name":"independent","page_name":"MirceaEinstein","display_name":"Mircea Einstein","profile_url":"https://independent.academia.edu/MirceaEinstein?f_ri=302","photo":"/images/s65_no_pic.png"}</script></span></span></li><li class="js-paper-rank-work_38300048 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="38300048"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 38300048, container: ".js-paper-rank-work_38300048", }); 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$(".js-view-count[data-work-id=38300048]").text(description); $(".js-view-count-work_38300048").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_38300048").removeClass('hidden') })</script></div></li><li class="InlineList-item u-positionRelative" style="max-width: 250px"><div class="u-positionAbsolute" data-has-card-for-ri-list="38300048"><i class="fa fa-tag InlineList-item-icon u-positionRelative"></i>  <a class="InlineList-item-text u-positionRelative">4</a>  </div><span class="InlineList-item-text u-textTruncate u-pl9x"><a class="InlineList-item-text" data-has-card-for-ri="301" rel="nofollow" href="https://www.academia.edu/Documents/in/Number_Theory">Number Theory</a>, <script data-card-contents-for-ri="301" type="text/json">{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=302","nofollow":true}</script><a class="InlineList-item-text" data-has-card-for-ri="302" rel="nofollow" href="https://www.academia.edu/Documents/in/Analytic_Number_Theory">Analytic Number Theory</a>, <script data-card-contents-for-ri="302" type="text/json">{"id":302,"name":"Analytic Number Theory","url":"https://www.academia.edu/Documents/in/Analytic_Number_Theory?f_ri=302","nofollow":true}</script><a class="InlineList-item-text" data-has-card-for-ri="370" rel="nofollow" href="https://www.academia.edu/Documents/in/Complex_Analysis">Complex Analysis</a>, <script data-card-contents-for-ri="370" type="text/json">{"id":370,"name":"Complex Analysis","url":"https://www.academia.edu/Documents/in/Complex_Analysis?f_ri=302","nofollow":true}</script><a class="InlineList-item-text" data-has-card-for-ri="256321" rel="nofollow" href="https://www.academia.edu/Documents/in/Prime_Number_Theory">Prime Number Theory</a><script data-card-contents-for-ri="256321" type="text/json">{"id":256321,"name":"Prime Number Theory","url":"https://www.academia.edu/Documents/in/Prime_Number_Theory?f_ri=302","nofollow":true}</script></span></li><script>(function(){ if (true) { new Aedu.ResearchInterestListCard({ el: $('*[data-has-card-for-ri-list=38300048]'), work: {"id":38300048,"title":"Distribution of primes, the computation of the non trivial zeros of the Riemann zeta function, root identities, replication identity, zero free zone, Weil conjecture and new mathematical frontier.","created_at":"2019-02-06T14:20:27.745-08:00","url":"https://www.academia.edu/38300048/Distribution_of_primes_the_computation_of_the_non_trivial_zeros_of_the_Riemann_zeta_function_root_identities_replication_identity_zero_free_zone_Weil_conjecture_and_new_mathematical_frontier?f_ri=302","dom_id":"work_38300048","summary":"In this paper we expand the prime number theorem, twice. Then we use both expansions to describe the distribution of primes. Afterwards we analyze the symmetry in the asymptotic equivalence. Before we come to the Weil conjecture, we derive a solution of the non-trivial zeros of the Riemann zeta function, the root identity, which we expand to the entire complex plane via a replication identity and which we employ to show, that the critical strip next to the critical line is a zero free zone. ","downloadable_attachments":[{"id":87783785,"asset_id":38300048,"asset_type":"Work","always_allow_download":false},{"id":82087924,"asset_id":38300048,"asset_type":"Work","always_allow_download":false},{"id":61138982,"asset_id":38300048,"asset_type":"Work","always_allow_download":false},{"id":105843092,"asset_id":38300048,"asset_type":"Work","always_allow_download":false},{"id":87522806,"asset_id":38300048,"asset_type":"Work","always_allow_download":false},{"id":94074483,"asset_id":38300048,"asset_type":"Work","always_allow_download":false},{"id":93360519,"asset_id":38300048,"asset_type":"Work","always_allow_download":false},{"id":93501888,"asset_id":38300048,"asset_type":"Work","always_allow_download":false},{"id":87520984,"asset_id":38300048,"asset_type":"Work","always_allow_download":false},{"id":87522239,"asset_id":38300048,"asset_type":"Work","always_allow_download":false},{"id":61124676,"asset_id":38300048,"asset_type":"Work","always_allow_download":false},{"id":104725494,"asset_id":38300048,"asset_type":"Work","always_allow_download":false},{"id":82090785,"asset_id":38300048,"asset_type":"Work","always_allow_download":false},{"id":105841936,"asset_id":38300048,"asset_type":"Work","always_allow_download":false},{"id":105844874,"asset_id":38300048,"asset_type":"Work","always_allow_download":false},{"id":82081316,"asset_id":38300048,"asset_type":"Work","always_allow_download":false},{"id":105844391,"asset_id":38300048,"asset_type":"Work","always_allow_download":false},{"id":87524219,"asset_id":38300048,"asset_type":"Work","always_allow_download":false},{"id":112400149,"asset_id":38300048,"asset_type":"Work","always_allow_download":false},{"id":112513341,"asset_id":38300048,"asset_type":"Work","always_allow_download":false},{"id":112513263,"asset_id":38300048,"asset_type":"Work","always_allow_download":false},{"id":112666385,"asset_id":38300048,"asset_type":"Work","always_allow_download":false},{"id":112731072,"asset_id":38300048,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":58174190,"first_name":"Mircea","last_name":"Einstein","domain_name":"independent","page_name":"MirceaEinstein","display_name":"Mircea Einstein","profile_url":"https://independent.academia.edu/MirceaEinstein?f_ri=302","photo":"/images/s65_no_pic.png"}],"research_interests":[{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=302","nofollow":true},{"id":302,"name":"Analytic Number Theory","url":"https://www.academia.edu/Documents/in/Analytic_Number_Theory?f_ri=302","nofollow":true},{"id":370,"name":"Complex Analysis","url":"https://www.academia.edu/Documents/in/Complex_Analysis?f_ri=302","nofollow":true},{"id":256321,"name":"Prime Number Theory","url":"https://www.academia.edu/Documents/in/Prime_Number_Theory?f_ri=302","nofollow":true}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_23485297" data-work_id="23485297" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/23485297/ANALYTIC_CONTINUATION_AND_FUNCTIONAL_EQUATION_OF_THE_DIRICHLET_ETA_FUNCTION">ANALYTIC CONTINUATION AND FUNCTIONAL EQUATION OF THE DIRICHLET ETA FUNCTION</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest">We shall show here a very simple way to obtain analytic continuation and functional equation for η(s) (and hence for ζ(s)) which is similar to that which was given by G. H. Hardy.</div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/23485297" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="1a9d80093674ecd78fe61a0fb1dcd382" rel="nofollow" data-download="{"attachment_id":43921603,"asset_id":23485297,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/43921603/download_file?st=MTc0MDg2MzQyOCw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="31961575" href="https://independent.academia.edu/emiliocarraro">emilio carraro</a><script data-card-contents-for-user="31961575" type="text/json">{"id":31961575,"first_name":"emilio","last_name":"carraro","domain_name":"independent","page_name":"emiliocarraro","display_name":"emilio carraro","profile_url":"https://independent.academia.edu/emiliocarraro?f_ri=302","photo":"/images/s65_no_pic.png"}</script></span></span></li><li class="js-paper-rank-work_23485297 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="23485297"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 23485297, container: ".js-paper-rank-work_23485297", }); });</script></li><li class="js-percentile-work_23485297 InlineList-item InlineList-item--bordered hidden u-tcGrayDark"><span class="percentile-widget hidden"><span class="u-mr2x percentile-widget" style="display: none">•</span><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 23485297; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-percentile-work_23485297"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></li><li class="js-view-count-work_23485297 InlineList-item InlineList-item--bordered hidden"><div><span><span class="js-view-count view-count u-mr2x" data-work-id="23485297"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 23485297; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=23485297]").text(description); $(".js-view-count-work_23485297").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_23485297").removeClass('hidden') })</script></div></li><li class="InlineList-item u-positionRelative" style="max-width: 250px"><div class="u-positionAbsolute" data-has-card-for-ri-list="23485297"><i class="fa fa-tag InlineList-item-icon u-positionRelative"></i></div><span class="InlineList-item-text u-textTruncate u-pl6x"><a class="InlineList-item-text" data-has-card-for-ri="302" rel="nofollow" href="https://www.academia.edu/Documents/in/Analytic_Number_Theory">Analytic Number Theory</a><script data-card-contents-for-ri="302" type="text/json">{"id":302,"name":"Analytic Number Theory","url":"https://www.academia.edu/Documents/in/Analytic_Number_Theory?f_ri=302","nofollow":true}</script></span></li><script>(function(){ if (false) { new Aedu.ResearchInterestListCard({ el: $('*[data-has-card-for-ri-list=23485297]'), work: {"id":23485297,"title":"ANALYTIC CONTINUATION AND FUNCTIONAL EQUATION OF THE DIRICHLET ETA FUNCTION","created_at":"2016-03-20T08:02:35.640-07:00","url":"https://www.academia.edu/23485297/ANALYTIC_CONTINUATION_AND_FUNCTIONAL_EQUATION_OF_THE_DIRICHLET_ETA_FUNCTION?f_ri=302","dom_id":"work_23485297","summary":"We shall show here a very simple way to obtain analytic continuation and functional equation for η(s) (and hence for ζ(s)) which is similar to that which was given by G. H. Hardy. ","downloadable_attachments":[{"id":43921603,"asset_id":23485297,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":31961575,"first_name":"emilio","last_name":"carraro","domain_name":"independent","page_name":"emiliocarraro","display_name":"emilio carraro","profile_url":"https://independent.academia.edu/emiliocarraro?f_ri=302","photo":"/images/s65_no_pic.png"}],"research_interests":[{"id":302,"name":"Analytic Number Theory","url":"https://www.academia.edu/Documents/in/Analytic_Number_Theory?f_ri=302","nofollow":true}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_71423580" data-work_id="71423580" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/71423580/A_Survey_Of_The_Riemann_Zeta_Function_With_Its_Applications">A Survey Of The Riemann Zeta Function With Its Applications</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest">This paper explicates the Riemann hypothesis and proves its validity. [The paper is published in a journal of number theory.]</div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/71423580" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="bb9c6d9101d3d21a374dbec07b897e60" rel="nofollow" data-download="{"attachment_id":80772877,"asset_id":71423580,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/80772877/download_file?st=MTc0MDg2MzQyOCw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="11278844" href="https://independent.academia.edu/BertrandWong">Bertrand Wong</a><script data-card-contents-for-user="11278844" type="text/json">{"id":11278844,"first_name":"Bertrand","last_name":"Wong","domain_name":"independent","page_name":"BertrandWong","display_name":"Bertrand Wong","profile_url":"https://independent.academia.edu/BertrandWong?f_ri=302","photo":"https://0.academia-photos.com/11278844/173241083/163252381/s65_bertrand.wong.png"}</script></span></span></li><li class="js-paper-rank-work_71423580 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="71423580"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 71423580, container: ".js-paper-rank-work_71423580", }); 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[The paper is published in a journal of number theory.]","downloadable_attachments":[{"id":80772877,"asset_id":71423580,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":11278844,"first_name":"Bertrand","last_name":"Wong","domain_name":"independent","page_name":"BertrandWong","display_name":"Bertrand Wong","profile_url":"https://independent.academia.edu/BertrandWong?f_ri=302","photo":"https://0.academia-photos.com/11278844/173241083/163252381/s65_bertrand.wong.png"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics?f_ri=302","nofollow":true},{"id":302,"name":"Analytic Number Theory","url":"https://www.academia.edu/Documents/in/Analytic_Number_Theory?f_ri=302","nofollow":true},{"id":303,"name":"Algebraic Number Theory","url":"https://www.academia.edu/Documents/in/Algebraic_Number_Theory?f_ri=302","nofollow":true},{"id":370,"name":"Complex 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mathematics","url":"https://www.academia.edu/Documents/in/Mesopotamian_mathematics?f_ri=302"},{"id":773824,"name":"Problem Posing in Mathematics","url":"https://www.academia.edu/Documents/in/Problem_Posing_in_Mathematics?f_ri=302"},{"id":904819,"name":"Using Digital Technology for Teaching Mathematics","url":"https://www.academia.edu/Documents/in/Using_Digital_Technology_for_Teaching_Mathematics?f_ri=302"}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_11284788" data-work_id="11284788" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/11284788/Golden_Ratio_in_the_Nature">Golden Ratio in the Nature</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest"><div class="summarized">Shapes defined by the golden ratio have long been considered aesthetically pleasing in western cultures, reflecting nature's balance between symmetry and asymmetry. The ratio is still used frequently in art and design. The golden ratio is... <a class="more_link u-tcGrayDark u-linkUnstyled" data-container=".work_11284788" data-show=".complete" data-hide=".summarized" data-more-link-behavior="true" href="#">more</a></div><div class="complete hidden">Shapes defined by the golden ratio have long been considered aesthetically pleasing in western cultures, reflecting nature's balance between symmetry and asymmetry. The ratio is still used frequently in art and design. The golden ratio is also known as the golden mean, golden section, golden number or divine proportion.It is usually denoted by the Greek letter  (phi).</div></div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/11284788" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="11d91ff7e325751e302c1a32188df6e9" rel="nofollow" data-download="{"attachment_id":36865048,"asset_id":11284788,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/36865048/download_file?st=MTc0MDg2MzQyOCw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="25839036" href="https://bamu.academia.edu/ashokgodase">Ashok D Godase</a><script data-card-contents-for-user="25839036" type="text/json">{"id":25839036,"first_name":"Ashok","last_name":"Godase","domain_name":"bamu","page_name":"ashokgodase","display_name":"Ashok D Godase","profile_url":"https://bamu.academia.edu/ashokgodase?f_ri=302","photo":"https://0.academia-photos.com/25839036/7699509/18732322/s65_ashok.godase.jpg"}</script></span></span></li><li class="js-paper-rank-work_11284788 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="11284788"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 11284788, container: ".js-paper-rank-work_11284788", }); 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The concept revolves around the relationship between different numerical bases... <a class="more_link u-tcGrayDark u-linkUnstyled" data-container=".work_2401099" data-show=".complete" data-hide=".summarized" data-more-link-behavior="true" href="#">more</a></div><div class="complete hidden">CHROMOSCALE presents a novel framework that integrates numerical systems with color representation through a unique chromatic approach based on cyclic orders. The concept revolves around the relationship between different numerical bases and their visual representations, specifically focusing on Base 7. The paper explores the mathematical properties of cyclic numbers, monotonic functions, and their implications for establishing a harmonic system that maps numbers to colors in a structured way, thereby creating a cohesive language for sounds, colors, and numbers. <br /> <br />For more information see the following link: <br /><a href="https://pdpseven.wixsite.com/sound-color" rel="nofollow">https://pdpseven.wixsite.com/sound-color</a></div></div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/2401099" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="1930006" href="https://independent.academia.edu/paolodipasquale">Paolo Di Pasquale</a><script data-card-contents-for-user="1930006" type="text/json">{"id":1930006,"first_name":"Paolo","last_name":"Di Pasquale","domain_name":"independent","page_name":"paolodipasquale","display_name":"Paolo Di Pasquale","profile_url":"https://independent.academia.edu/paolodipasquale?f_ri=302","photo":"https://0.academia-photos.com/1930006/651321/807219/s65_paolo.di_pasquale.jpg"}</script></span></span></li><li class="js-paper-rank-work_2401099 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="2401099"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 2401099, container: ".js-paper-rank-work_2401099", }); 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on the Riemann zeta function. Most of the core ideas of that paper, of which a majority are due to... <a class="more_link u-tcGrayDark u-linkUnstyled" data-container=".work_3221296" data-show=".complete" data-hide=".summarized" data-more-link-behavior="true" href="#">more</a></div><div class="complete hidden">This paper consists of the extended working notes and observations made during the development of a joint paper [?] with Philippe Flajolet on the Riemann zeta function. Most of the core ideas of that paper, of which a majority are due to Flajolet, are reproduced here; however, the choice of wording used here, and all errors and omissions are my own fault. This set of notes contains considerably more content, although is looser and sloppier, and is an exploration of tangents, dead-ends, and ideas shooting off in uncertain directions.</div></div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/3221296" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="9fa62a488a131dbd03216213821aeebe" rel="nofollow" data-download="{"attachment_id":31090808,"asset_id":3221296,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/31090808/download_file?st=MTc0MDg2MzQyOCw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="3685531" href="https://independent.academia.edu/LinasVepstas">Linas Vepstas</a><script data-card-contents-for-user="3685531" type="text/json">{"id":3685531,"first_name":"Linas","last_name":"Vepstas","domain_name":"independent","page_name":"LinasVepstas","display_name":"Linas Vepstas","profile_url":"https://independent.academia.edu/LinasVepstas?f_ri=302","photo":"https://gravatar.com/avatar/a221b4b4e6dd199750731c51ee864a91?s=65"}</script></span></span></li><li class="js-paper-rank-work_3221296 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="3221296"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 3221296, container: ".js-paper-rank-work_3221296", }); 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$(".js-view-count[data-work-id=3221296]").text(description); $(".js-view-count-work_3221296").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_3221296").removeClass('hidden') })</script></div></li><li class="InlineList-item u-positionRelative" style="max-width: 250px"><div class="u-positionAbsolute" data-has-card-for-ri-list="3221296"><i class="fa fa-tag InlineList-item-icon u-positionRelative"></i>  <a class="InlineList-item-text u-positionRelative">2</a>  </div><span class="InlineList-item-text u-textTruncate u-pl9x"><a class="InlineList-item-text" data-has-card-for-ri="301" rel="nofollow" href="https://www.academia.edu/Documents/in/Number_Theory">Number Theory</a>, <script data-card-contents-for-ri="301" type="text/json">{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=302","nofollow":true}</script><a class="InlineList-item-text" data-has-card-for-ri="302" rel="nofollow" href="https://www.academia.edu/Documents/in/Analytic_Number_Theory">Analytic Number Theory</a><script data-card-contents-for-ri="302" type="text/json">{"id":302,"name":"Analytic Number Theory","url":"https://www.academia.edu/Documents/in/Analytic_Number_Theory?f_ri=302","nofollow":true}</script></span></li><script>(function(){ if (true) { new Aedu.ResearchInterestListCard({ el: $('*[data-has-card-for-ri-list=3221296]'), work: {"id":3221296,"title":"Notes relating to newton series for the riemann zeta function","created_at":"2013-04-05T14:05:54.885-07:00","url":"https://www.academia.edu/3221296/Notes_relating_to_newton_series_for_the_riemann_zeta_function?f_ri=302","dom_id":"work_3221296","summary":"This paper consists of the extended working notes and observations made during the development of a joint paper [?] with Philippe Flajolet on the Riemann zeta function. Most of the core ideas of that paper, of which a majority are due to Flajolet, are reproduced here; however, the choice of wording used here, and all errors and omissions are my own fault. This set of notes contains considerably more content, although is looser and sloppier, and is an exploration of tangents, dead-ends, and ideas shooting off in uncertain directions.","downloadable_attachments":[{"id":31090808,"asset_id":3221296,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":3685531,"first_name":"Linas","last_name":"Vepstas","domain_name":"independent","page_name":"LinasVepstas","display_name":"Linas Vepstas","profile_url":"https://independent.academia.edu/LinasVepstas?f_ri=302","photo":"https://gravatar.com/avatar/a221b4b4e6dd199750731c51ee864a91?s=65"}],"research_interests":[{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=302","nofollow":true},{"id":302,"name":"Analytic Number Theory","url":"https://www.academia.edu/Documents/in/Analytic_Number_Theory?f_ri=302","nofollow":true}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_9343239" data-work_id="9343239" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/9343239/Re_conceptualizing_Information_System_Success_The_IS_Impact_Measurement_Model_Re_conceptualizing_Information_System_Success_The_IS_Impact_Measurement_Model">Re-conceptualizing Information System Success: The IS-Impact Measurement Model * Re-conceptualizing Information System Success: The IS-Impact Measurement Model</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest"><div class="summarized">This paper re-conceptualizes "information system success" as a formative, multidimensional index. Such a validated and widely accepted index would facilitate cumulative research on the impacts of IS, while at the same time provide a... <a class="more_link u-tcGrayDark u-linkUnstyled" data-container=".work_9343239" data-show=".complete" data-hide=".summarized" data-more-link-behavior="true" href="#">more</a></div><div class="complete hidden">This paper re-conceptualizes "information system success" as a formative, multidimensional index. Such a validated and widely accepted index would facilitate cumulative research on the impacts of IS, while at the same time provide a benchmark for organizations to track their IS performance. The proposed IS-Impact measurement model represents the stream of net benefits from an Information System (IS), to date and anticipated, as perceived by all key user groups. Model measures are formulated to be robust, economical, and simple, yielding results that are comparable across diverse systems and contexts, and from multiple user perspectives. The model includes four dimensions in two halves. The "impact" half measures benefits to date, or Individualand Organizational Impact; the "quality" half uses System Quality and Information Quality as proxies for probable future impacts. Study findings evidence the necessity, additivity, and completeness of these four dimensions.</div></div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/9343239" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="95f1c94b6cf8374a28d145f6dbd1518a" rel="nofollow" data-download="{"attachment_id":35599296,"asset_id":9343239,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/35599296/download_file?st=MTc0MDg2MzQyOCw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="21614075" href="https://independent.academia.edu/NewTurtleGym">New Turtle Gym</a><script data-card-contents-for-user="21614075" type="text/json">{"id":21614075,"first_name":"New Turtle","last_name":"Gym","domain_name":"independent","page_name":"NewTurtleGym","display_name":"New Turtle Gym","profile_url":"https://independent.academia.edu/NewTurtleGym?f_ri=302","photo":"/images/s65_no_pic.png"}</script></span></span></li><li class="js-paper-rank-work_9343239 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="9343239"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 9343239, container: ".js-paper-rank-work_9343239", }); 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$(".js-view-count[data-work-id=9343239]").text(description); $(".js-view-count-work_9343239").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_9343239").removeClass('hidden') })</script></div></li><li class="InlineList-item u-positionRelative" style="max-width: 250px"><div class="u-positionAbsolute" data-has-card-for-ri-list="9343239"><i class="fa fa-tag InlineList-item-icon u-positionRelative"></i>  <a class="InlineList-item-text u-positionRelative">6</a>  </div><span class="InlineList-item-text u-textTruncate u-pl9x"><a class="InlineList-item-text" data-has-card-for-ri="302" rel="nofollow" href="https://www.academia.edu/Documents/in/Analytic_Number_Theory">Analytic Number Theory</a>, <script data-card-contents-for-ri="302" type="text/json">{"id":302,"name":"Analytic Number Theory","url":"https://www.academia.edu/Documents/in/Analytic_Number_Theory?f_ri=302","nofollow":true}</script><a class="InlineList-item-text" data-has-card-for-ri="13886" rel="nofollow" href="https://www.academia.edu/Documents/in/Longitudinal_Research">Longitudinal Research</a>, <script data-card-contents-for-ri="13886" type="text/json">{"id":13886,"name":"Longitudinal Research","url":"https://www.academia.edu/Documents/in/Longitudinal_Research?f_ri=302","nofollow":true}</script><a class="InlineList-item-text" data-has-card-for-ri="232084" rel="nofollow" href="https://www.academia.edu/Documents/in/IS_impact">IS impact</a>, <script data-card-contents-for-ri="232084" type="text/json">{"id":232084,"name":"IS impact","url":"https://www.academia.edu/Documents/in/IS_impact?f_ri=302","nofollow":true}</script><a class="InlineList-item-text" data-has-card-for-ri="335153" rel="nofollow" href="https://www.academia.edu/Documents/in/Enterprise_System">Enterprise System</a><script data-card-contents-for-ri="335153" type="text/json">{"id":335153,"name":"Enterprise System","url":"https://www.academia.edu/Documents/in/Enterprise_System?f_ri=302","nofollow":true}</script></span></li><script>(function(){ if (true) { new Aedu.ResearchInterestListCard({ el: $('*[data-has-card-for-ri-list=9343239]'), work: {"id":9343239,"title":"Re-conceptualizing Information System Success: The IS-Impact Measurement Model * Re-conceptualizing Information System Success: The IS-Impact Measurement Model","created_at":"2014-11-16T18:19:09.604-08:00","url":"https://www.academia.edu/9343239/Re_conceptualizing_Information_System_Success_The_IS_Impact_Measurement_Model_Re_conceptualizing_Information_System_Success_The_IS_Impact_Measurement_Model?f_ri=302","dom_id":"work_9343239","summary":"This paper re-conceptualizes \"information system success\" as a formative, multidimensional index. Such a validated and widely accepted index would facilitate cumulative research on the impacts of IS, while at the same time provide a benchmark for organizations to track their IS performance. The proposed IS-Impact measurement model represents the stream of net benefits from an Information System (IS), to date and anticipated, as perceived by all key user groups. Model measures are formulated to be robust, economical, and simple, yielding results that are comparable across diverse systems and contexts, and from multiple user perspectives. The model includes four dimensions in two halves. The \"impact\" half measures benefits to date, or Individualand Organizational Impact; the \"quality\" half uses System Quality and Information Quality as proxies for probable future impacts. Study findings evidence the necessity, additivity, and completeness of these four dimensions.","downloadable_attachments":[{"id":35599296,"asset_id":9343239,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":21614075,"first_name":"New Turtle","last_name":"Gym","domain_name":"independent","page_name":"NewTurtleGym","display_name":"New Turtle Gym","profile_url":"https://independent.academia.edu/NewTurtleGym?f_ri=302","photo":"/images/s65_no_pic.png"}],"research_interests":[{"id":302,"name":"Analytic Number Theory","url":"https://www.academia.edu/Documents/in/Analytic_Number_Theory?f_ri=302","nofollow":true},{"id":13886,"name":"Longitudinal Research","url":"https://www.academia.edu/Documents/in/Longitudinal_Research?f_ri=302","nofollow":true},{"id":232084,"name":"IS impact","url":"https://www.academia.edu/Documents/in/IS_impact?f_ri=302","nofollow":true},{"id":335153,"name":"Enterprise System","url":"https://www.academia.edu/Documents/in/Enterprise_System?f_ri=302","nofollow":true},{"id":382786,"name":"Measurement Model","url":"https://www.academia.edu/Documents/in/Measurement_Model?f_ri=302"},{"id":1457067,"name":"Questionnaire Survey","url":"https://www.academia.edu/Documents/in/Questionnaire_Survey?f_ri=302"}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_2277636" data-work_id="2277636" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/2277636/The_Harmonic_Math_Chromatic_numbers_and_sounds">The Harmonic Math, Chromatic numbers and sounds</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest"><div class="summarized">"The Harmonic Math, Chromatic numbers and sounds" presents a novel framework that integrates numerical systems with color representation through a unique chromatic approach based on cyclic orders. The concept revolves around the... <a class="more_link u-tcGrayDark u-linkUnstyled" data-container=".work_2277636" data-show=".complete" data-hide=".summarized" data-more-link-behavior="true" href="#">more</a></div><div class="complete hidden">"The Harmonic Math, Chromatic numbers and sounds" presents a novel framework that integrates numerical systems with color representation through a unique chromatic approach based on cyclic orders. The concept revolves around the relationship between different numerical bases and their visual representations, specifically focusing on Base 7. The paper explores the mathematical properties of cyclic numbers, monotonic functions, and their implications for establishing a harmonic system that maps numbers to colors in a structured way, thereby creating a cohesive language for sounds, colors, and numbers. <br />For more information see the following link: <a href="https://pdpseven.wixsite.com/sound-color" rel="nofollow">https://pdpseven.wixsite.com/sound-color</a></div></div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/2277636" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="1930006" href="https://independent.academia.edu/paolodipasquale">Paolo Di Pasquale</a><script data-card-contents-for-user="1930006" type="text/json">{"id":1930006,"first_name":"Paolo","last_name":"Di Pasquale","domain_name":"independent","page_name":"paolodipasquale","display_name":"Paolo Di Pasquale","profile_url":"https://independent.academia.edu/paolodipasquale?f_ri=302","photo":"https://0.academia-photos.com/1930006/651321/807219/s65_paolo.di_pasquale.jpg"}</script></span></span></li><li class="js-paper-rank-work_2277636 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="2277636"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 2277636, container: ".js-paper-rank-work_2277636", }); 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$(".js-view-count[data-work-id=2277636]").text(description); $(".js-view-count-work_2277636").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_2277636").removeClass('hidden') })</script></div></li><li class="InlineList-item u-positionRelative" style="max-width: 250px"><div class="u-positionAbsolute" data-has-card-for-ri-list="2277636"><i class="fa fa-tag InlineList-item-icon u-positionRelative"></i>  <a class="InlineList-item-text u-positionRelative">385</a>  </div><span class="InlineList-item-text u-textTruncate u-pl11x"><a class="InlineList-item-text" data-has-card-for-ri="1" rel="nofollow" href="https://www.academia.edu/Documents/in/Creative_Writing">Creative Writing</a>, <script data-card-contents-for-ri="1" type="text/json">{"id":1,"name":"Creative Writing","url":"https://www.academia.edu/Documents/in/Creative_Writing?f_ri=302","nofollow":true}</script><a class="InlineList-item-text" data-has-card-for-ri="1" rel="nofollow" href="https://www.academia.edu/Documents/in/Creative_Writing">Creative Writing</a>, <script data-card-contents-for-ri="1" type="text/json">{"id":1,"name":"Creative Writing","url":"https://www.academia.edu/Documents/in/Creative_Writing?f_ri=302","nofollow":true}</script><a class="InlineList-item-text" data-has-card-for-ri="8" rel="nofollow" href="https://www.academia.edu/Documents/in/Critical_Theory">Critical Theory</a>, <script data-card-contents-for-ri="8" type="text/json">{"id":8,"name":"Critical Theory","url":"https://www.academia.edu/Documents/in/Critical_Theory?f_ri=302","nofollow":true}</script><a class="InlineList-item-text" data-has-card-for-ri="106" rel="nofollow" href="https://www.academia.edu/Documents/in/Religion">Religion</a><script data-card-contents-for-ri="106" type="text/json">{"id":106,"name":"Religion","url":"https://www.academia.edu/Documents/in/Religion?f_ri=302","nofollow":true}</script></span></li><script>(function(){ if (true) { new Aedu.ResearchInterestListCard({ el: $('*[data-has-card-for-ri-list=2277636]'), work: {"id":2277636,"title":"The Harmonic Math, Chromatic numbers and sounds","created_at":"2012-12-11T06:30:23.436-08:00","url":"https://www.academia.edu/2277636/The_Harmonic_Math_Chromatic_numbers_and_sounds?f_ri=302","dom_id":"work_2277636","summary":"\"The Harmonic Math, Chromatic numbers and sounds\" presents a novel framework that integrates numerical systems with color representation through a unique chromatic approach based on cyclic orders. The concept revolves around the relationship between different numerical bases and their visual representations, specifically focusing on Base 7. The paper explores the mathematical properties of cyclic numbers, monotonic functions, and their implications for establishing a harmonic system that maps numbers to colors in a structured way, thereby creating a cohesive language for sounds, colors, and numbers. \r\nFor more information see the following link: https://pdpseven.wixsite.com/sound-color","downloadable_attachments":[],"ordered_authors":[{"id":1930006,"first_name":"Paolo","last_name":"Di Pasquale","domain_name":"independent","page_name":"paolodipasquale","display_name":"Paolo Di Pasquale","profile_url":"https://independent.academia.edu/paolodipasquale?f_ri=302","photo":"https://0.academia-photos.com/1930006/651321/807219/s65_paolo.di_pasquale.jpg"}],"research_interests":[{"id":1,"name":"Creative Writing","url":"https://www.academia.edu/Documents/in/Creative_Writing?f_ri=302","nofollow":true},{"id":1,"name":"Creative 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Zeta Function</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest"><div class="summarized">This paper shows why the non-trivial zeros of the Riemann zeta function ζ must always be on the critical line Re(s) = 1/2 and not anywhere else on the critical strip bounded by Re(s) = 0 and Re(s) = 1, thus affirming the validity of the... <a class="more_link u-tcGrayDark u-linkUnstyled" data-container=".work_71423610" data-show=".complete" data-hide=".summarized" data-more-link-behavior="true" href="#">more</a></div><div class="complete hidden">This paper shows why the non-trivial zeros of the Riemann zeta function ζ must always be on the critical line Re(s) = 1/2 and not anywhere else on the critical strip bounded by Re(s) = 0 and Re(s) = 1, thus affirming the validity of the Riemann hypothesis. 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Why these three P? I must confess that the idea of trying to define the analysis activity, and implicitly, the main parts of a reform in this important area for the activity of each intelligence service came following some discussion I had with colleagues, experts and intelligence analysts. I worked, as co-author, to a paper (presented within an international forum) concerning the evolution of intelligenc...</div></div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/81963448" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="ddf34b9dbbe8a219781ae201030f75e2" rel="nofollow" data-download="{"attachment_id":87823791,"asset_id":81963448,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/87823791/download_file?st=MTc0MDg2MzQyOCw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="30436646" href="https://independent.academia.edu/IonelNitu">Ionel Nitu</a><script data-card-contents-for-user="30436646" type="text/json">{"id":30436646,"first_name":"Ionel","last_name":"Nitu","domain_name":"independent","page_name":"IonelNitu","display_name":"Ionel Nitu","profile_url":"https://independent.academia.edu/IonelNitu?f_ri=302","photo":"https://0.academia-photos.com/30436646/8833268/9861472/s65_ionel.nitu.jpg"}</script></span></span></li><li class="js-paper-rank-work_81963448 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="81963448"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 81963448, container: ".js-paper-rank-work_81963448", }); 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– Product (the results of the analysis activity, the p roducts which are sent to beneficiaries / users and the feedback or the requests for inform ation from the intelligence consumers); – Personnel (the intelligence analyst, as well as the process of its selection and training). Why these three P? I must confess that the idea of trying to define the analysis activity, and implicitly, the main parts of a reform in this important area for the activity of each intelligence service came following some discussion I had with colleagues, experts and intelligence analysts. I worked, as co-author, to a paper (presented within an international forum) concerning the evolution of intelligenc...","downloadable_attachments":[{"id":87823791,"asset_id":81963448,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":30436646,"first_name":"Ionel","last_name":"Nitu","domain_name":"independent","page_name":"IonelNitu","display_name":"Ionel Nitu","profile_url":"https://independent.academia.edu/IonelNitu?f_ri=302","photo":"https://0.academia-photos.com/30436646/8833268/9861472/s65_ionel.nitu.jpg"}],"research_interests":[{"id":184,"name":"Sociology","url":"https://www.academia.edu/Documents/in/Sociology?f_ri=302","nofollow":true},{"id":302,"name":"Analytic Number Theory","url":"https://www.academia.edu/Documents/in/Analytic_Number_Theory?f_ri=302","nofollow":true},{"id":2674,"name":"Intelligence","url":"https://www.academia.edu/Documents/in/Intelligence?f_ri=302","nofollow":true},{"id":4486,"name":"Political Science","url":"https://www.academia.edu/Documents/in/Political_Science?f_ri=302","nofollow":true},{"id":6662,"name":"Secrecy","url":"https://www.academia.edu/Documents/in/Secrecy?f_ri=302"},{"id":48952,"name":"Romania","url":"https://www.academia.edu/Documents/in/Romania?f_ri=302"},{"id":52604,"name":"Recruitment","url":"https://www.academia.edu/Documents/in/Recruitment?f_ri=302"},{"id":61620,"name":"National Security","url":"https://www.academia.edu/Documents/in/National_Security?f_ri=302"},{"id":94440,"name":"Process management","url":"https://www.academia.edu/Documents/in/Process_management?f_ri=302"},{"id":129624,"name":"Organizational Sociology","url":"https://www.academia.edu/Documents/in/Organizational_Sociology?f_ri=302"},{"id":257915,"name":"Misc","url":"https://www.academia.edu/Documents/in/Misc?f_ri=302"},{"id":287043,"name":"Manpower","url":"https://www.academia.edu/Documents/in/Manpower?f_ri=302"},{"id":403784,"name":"Politikwissenschaft","url":"https://www.academia.edu/Documents/in/Politikwissenschaft?f_ri=302"},{"id":799504,"name":"Organizational Analysis","url":"https://www.academia.edu/Documents/in/Organizational_Analysis?f_ri=302"},{"id":1019146,"name":"Visual Anthropology and Sociology","url":"https://www.academia.edu/Documents/in/Visual_Anthropology_and_Sociology?f_ri=302"},{"id":1028001,"name":"Intelligenz","url":"https://www.academia.edu/Documents/in/Intelligenz?f_ri=302"},{"id":1929976,"name":"Rumänien","url":"https://www.academia.edu/Documents/in/Rum%C3%A4nien?f_ri=302"},{"id":2469006,"name":"Secret service","url":"https://www.academia.edu/Documents/in/Secret_service?f_ri=302"},{"id":2989544,"name":"Organisationsanalyse","url":"https://www.academia.edu/Documents/in/Organisationsanalyse?f_ri=302"},{"id":4045092,"name":"Geheimdienst","url":"https://www.academia.edu/Documents/in/Geheimdienst?f_ri=302"}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_3221295" data-work_id="3221295" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/3221295/An_efficient_algorithm_for_accelerating_the_convergence_of_oscillatory_series_useful_for_computing_the_polylogarithm_and_Hurwitz_zeta_functions">An efficient algorithm for accelerating the convergence of oscillatory series, useful for computing the polylogarithm and Hurwitz zeta functions</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest"><div class="summarized">This paper sketches a technique for improving the rate of convergence of a general oscillatory sequence, and then applies this series acceleration algorithm to the polylogarithm and the Hurwitz zeta function. As such, it may be taken as... <a class="more_link u-tcGrayDark u-linkUnstyled" data-container=".work_3221295" data-show=".complete" data-hide=".summarized" data-more-link-behavior="true" href="#">more</a></div><div class="complete hidden">This paper sketches a technique for improving the rate of convergence of a general oscillatory sequence, and then applies this series acceleration algorithm to the polylogarithm and the Hurwitz zeta function. As such, it may be taken as an extension of the techniques given by Borwein's "An efficient algorithm for computing the Riemann zeta function" 5], to more general series. The algorithm provides a rapid means of evaluating Li s (z) for general values of complex s and a kidney-shaped region of complex z values given by z 2 /(z − 1) < 4. By using the duplication formula and the inversion formula, the range of convergence for the polylogarithm may be extended to the entire complex z-plane, and so the algorithms described here allow for the evaluation of the polylogarithm for all complex s and z values.</div></div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/3221295" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="5d548ad54e3abe7bcf3bd85ae9b0f752" rel="nofollow" data-download="{"attachment_id":50407191,"asset_id":3221295,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/50407191/download_file?st=MTc0MDg2MzQyOCw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="3685531" href="https://independent.academia.edu/LinasVepstas">Linas Vepstas</a><script data-card-contents-for-user="3685531" type="text/json">{"id":3685531,"first_name":"Linas","last_name":"Vepstas","domain_name":"independent","page_name":"LinasVepstas","display_name":"Linas Vepstas","profile_url":"https://independent.academia.edu/LinasVepstas?f_ri=302","photo":"https://gravatar.com/avatar/a221b4b4e6dd199750731c51ee864a91?s=65"}</script></span></span></li><li class="js-paper-rank-work_3221295 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="3221295"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 3221295, container: ".js-paper-rank-work_3221295", }); 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$(".js-view-count[data-work-id=3221295]").text(description); $(".js-view-count-work_3221295").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_3221295").removeClass('hidden') })</script></div></li><li class="InlineList-item u-positionRelative" style="max-width: 250px"><div class="u-positionAbsolute" data-has-card-for-ri-list="3221295"><i class="fa fa-tag InlineList-item-icon u-positionRelative"></i>  <a class="InlineList-item-text u-positionRelative">2</a>  </div><span class="InlineList-item-text u-textTruncate u-pl9x"><a class="InlineList-item-text" data-has-card-for-ri="302" rel="nofollow" href="https://www.academia.edu/Documents/in/Analytic_Number_Theory">Analytic Number Theory</a>, <script data-card-contents-for-ri="302" type="text/json">{"id":302,"name":"Analytic Number Theory","url":"https://www.academia.edu/Documents/in/Analytic_Number_Theory?f_ri=302","nofollow":true}</script><a class="InlineList-item-text" data-has-card-for-ri="706253" rel="nofollow" href="https://www.academia.edu/Documents/in/Polylogarithms">Polylogarithms</a><script data-card-contents-for-ri="706253" type="text/json">{"id":706253,"name":"Polylogarithms","url":"https://www.academia.edu/Documents/in/Polylogarithms?f_ri=302","nofollow":true}</script></span></li><script>(function(){ if (true) { new Aedu.ResearchInterestListCard({ el: $('*[data-has-card-for-ri-list=3221295]'), work: {"id":3221295,"title":"An efficient algorithm for accelerating the convergence of oscillatory series, useful for computing the polylogarithm and Hurwitz zeta functions","created_at":"2013-04-05T14:05:54.116-07:00","url":"https://www.academia.edu/3221295/An_efficient_algorithm_for_accelerating_the_convergence_of_oscillatory_series_useful_for_computing_the_polylogarithm_and_Hurwitz_zeta_functions?f_ri=302","dom_id":"work_3221295","summary":"This paper sketches a technique for improving the rate of convergence of a general oscillatory sequence, and then applies this series acceleration algorithm to the polylogarithm and the Hurwitz zeta function. As such, it may be taken as an extension of the techniques given by Borwein's \"An efficient algorithm for computing the Riemann zeta function\" 5], to more general series. The algorithm provides a rapid means of evaluating Li s (z) for general values of complex s and a kidney-shaped region of complex z values given by z 2 /(z − 1) \u003c 4. By using the duplication formula and the inversion formula, the range of convergence for the polylogarithm may be extended to the entire complex z-plane, and so the algorithms described here allow for the evaluation of the polylogarithm for all complex s and z values.","downloadable_attachments":[{"id":50407191,"asset_id":3221295,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":3685531,"first_name":"Linas","last_name":"Vepstas","domain_name":"independent","page_name":"LinasVepstas","display_name":"Linas Vepstas","profile_url":"https://independent.academia.edu/LinasVepstas?f_ri=302","photo":"https://gravatar.com/avatar/a221b4b4e6dd199750731c51ee864a91?s=65"}],"research_interests":[{"id":302,"name":"Analytic Number Theory","url":"https://www.academia.edu/Documents/in/Analytic_Number_Theory?f_ri=302","nofollow":true},{"id":706253,"name":"Polylogarithms","url":"https://www.academia.edu/Documents/in/Polylogarithms?f_ri=302","nofollow":true}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_81782485" data-work_id="81782485" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/81782485/Non_Trivial_Zeros_of_Riemann_Zeta_Function_and_Riemann_Hypothesis">Non-Trivial Zeros of Riemann Zeta Function and Riemann Hypothesis</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest"><div class="summarized">This paper, which is published in an international mathematics journal and endorsed by a mathematics society, touches on the part played by the non-trivial zeros of the Riemann zeta function ζ, providing many important information and... <a class="more_link u-tcGrayDark u-linkUnstyled" data-container=".work_81782485" data-show=".complete" data-hide=".summarized" data-more-link-behavior="true" href="#">more</a></div><div class="complete hidden">This paper, which is published in an international mathematics journal and endorsed by a mathematics society, touches on the part played by the non-trivial zeros of the Riemann zeta function ζ, providing many important information and insights in the process, including some approaches to the Riemann hypothesis.</div></div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/81782485" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="ef85691223842737595031ada0be4147" rel="nofollow" data-download="{"attachment_id":88052281,"asset_id":81782485,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/88052281/download_file?st=MTc0MDg2MzQyOCw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="11414130" href="https://independent.academia.edu/WongBertrand">Bertrand Wong</a><script data-card-contents-for-user="11414130" type="text/json">{"id":11414130,"first_name":"Bertrand","last_name":"Wong","domain_name":"independent","page_name":"WongBertrand","display_name":"Bertrand Wong","profile_url":"https://independent.academia.edu/WongBertrand?f_ri=302","photo":"https://0.academia-photos.com/11414130/3329015/3917133/s65_bertrand.wong.jpg"}</script></span></span></li><li class="js-paper-rank-work_81782485 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="81782485"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 81782485, container: ".js-paper-rank-work_81782485", }); 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$(".js-view-count[data-work-id=81782485]").text(description); $(".js-view-count-work_81782485").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_81782485").removeClass('hidden') })</script></div></li><li class="InlineList-item u-positionRelative" style="max-width: 250px"><div class="u-positionAbsolute" data-has-card-for-ri-list="81782485"><i class="fa fa-tag InlineList-item-icon u-positionRelative"></i>  <a class="InlineList-item-text u-positionRelative">19</a>  </div><span class="InlineList-item-text u-textTruncate u-pl10x"><a class="InlineList-item-text" data-has-card-for-ri="300" rel="nofollow" href="https://www.academia.edu/Documents/in/Mathematics">Mathematics</a>, <script data-card-contents-for-ri="300" type="text/json">{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics?f_ri=302","nofollow":true}</script><a class="InlineList-item-text" data-has-card-for-ri="301" rel="nofollow" href="https://www.academia.edu/Documents/in/Number_Theory">Number Theory</a>, <script data-card-contents-for-ri="301" type="text/json">{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=302","nofollow":true}</script><a class="InlineList-item-text" data-has-card-for-ri="302" rel="nofollow" href="https://www.academia.edu/Documents/in/Analytic_Number_Theory">Analytic Number Theory</a>, <script data-card-contents-for-ri="302" type="text/json">{"id":302,"name":"Analytic Number Theory","url":"https://www.academia.edu/Documents/in/Analytic_Number_Theory?f_ri=302","nofollow":true}</script><a class="InlineList-item-text" data-has-card-for-ri="369" rel="nofollow" href="https://www.academia.edu/Documents/in/Calculus">Calculus</a><script data-card-contents-for-ri="369" type="text/json">{"id":369,"name":"Calculus","url":"https://www.academia.edu/Documents/in/Calculus?f_ri=302","nofollow":true}</script></span></li><script>(function(){ if (true) { new Aedu.ResearchInterestListCard({ el: $('*[data-has-card-for-ri-list=81782485]'), work: {"id":81782485,"title":"Non-Trivial Zeros of Riemann Zeta Function and Riemann Hypothesis","created_at":"2022-06-18T13:43:40.550-07:00","url":"https://www.academia.edu/81782485/Non_Trivial_Zeros_of_Riemann_Zeta_Function_and_Riemann_Hypothesis?f_ri=302","dom_id":"work_81782485","summary":"This paper, which is published in an international mathematics journal and endorsed by a mathematics society, touches on the part played by the non-trivial zeros of the Riemann zeta function ζ, providing many important information and insights in the process, including some approaches to the Riemann hypothesis. ","downloadable_attachments":[{"id":88052281,"asset_id":81782485,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":11414130,"first_name":"Bertrand","last_name":"Wong","domain_name":"independent","page_name":"WongBertrand","display_name":"Bertrand Wong","profile_url":"https://independent.academia.edu/WongBertrand?f_ri=302","photo":"https://0.academia-photos.com/11414130/3329015/3917133/s65_bertrand.wong.jpg"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics?f_ri=302","nofollow":true},{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=302","nofollow":true},{"id":302,"name":"Analytic Number Theory","url":"https://www.academia.edu/Documents/in/Analytic_Number_Theory?f_ri=302","nofollow":true},{"id":369,"name":"Calculus","url":"https://www.academia.edu/Documents/in/Calculus?f_ri=302","nofollow":true},{"id":370,"name":"Complex Analysis","url":"https://www.academia.edu/Documents/in/Complex_Analysis?f_ri=302"},{"id":422,"name":"Computer Science","url":"https://www.academia.edu/Documents/in/Computer_Science?f_ri=302"},{"id":2509,"name":"Philosophy Of Mathematics","url":"https://www.academia.edu/Documents/in/Philosophy_Of_Mathematics?f_ri=302"},{"id":19997,"name":"Pure Mathematics","url":"https://www.academia.edu/Documents/in/Pure_Mathematics?f_ri=302"},{"id":20254,"name":"Elementary Number Theory","url":"https://www.academia.edu/Documents/in/Elementary_Number_Theory?f_ri=302"},{"id":49823,"name":"Distribution of Prime Numbers","url":"https://www.academia.edu/Documents/in/Distribution_of_Prime_Numbers?f_ri=302"},{"id":72443,"name":"Prime Numbers","url":"https://www.academia.edu/Documents/in/Prime_Numbers?f_ri=302"},{"id":168994,"name":"Functional Analysis, Mathematical Analysis and its applications","url":"https://www.academia.edu/Documents/in/Functional_Analysis_Mathematical_Analysis_and_its_applications?f_ri=302"},{"id":229028,"name":"riemann Hypothesis","url":"https://www.academia.edu/Documents/in/riemann_Hypothesis?f_ri=302"},{"id":256321,"name":"Prime Number Theory","url":"https://www.academia.edu/Documents/in/Prime_Number_Theory?f_ri=302"},{"id":459810,"name":"Differential Calculus","url":"https://www.academia.edu/Documents/in/Differential_Calculus?f_ri=302"},{"id":466067,"name":"Complex Numbers","url":"https://www.academia.edu/Documents/in/Complex_Numbers?f_ri=302"},{"id":615740,"name":"Functions of a complex variable","url":"https://www.academia.edu/Documents/in/Functions_of_a_complex_variable?f_ri=302"},{"id":968248,"name":"Fermat's Last Theorem","url":"https://www.academia.edu/Documents/in/Fermats_Last_Theorem?f_ri=302"},{"id":1149271,"name":"Integral Calculus","url":"https://www.academia.edu/Documents/in/Integral_Calculus?f_ri=302"}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_31106546" data-work_id="31106546" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/31106546/Number_Theory_in_Byzantium_according_to_Codex_Vindobonensis_phil_Graecus_65_of_the_15th_cent_The_numbers_and_the_numeric_position_in_Byzantium_BULLETIN_OF_THE_HELLENIC_MATHEMATICAL_SOCIETY_Volume_59_2016_107_118_Announcement_at_INTERNATIONAL_CONGRESS_ON_MATHEMATICS_MICOM_2015">''Number Theory in Byzantium according to Codex Vindobonensis phil. Graecus 65 of the 15th cent The numbers and the numeric position in Byzantium''. BULLETIN OF THE HELLENIC MATHEMATICAL SOCIETY Volume 59, 2016 (107-118). Announcement at INTERNATIONAL CONGRESS ON MATHEMATICS, MICOM 2015</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest"><div class="summarized">In this article we describe some elements from the chapter of Mathematics " Number Theory " , as they appear in the Codex Vindobonensis phil. Gr. 65, a Byzantine Ms kept in the National Library of Austria in Vienna. This codex contains a... <a class="more_link u-tcGrayDark u-linkUnstyled" data-container=".work_31106546" data-show=".complete" data-hide=".summarized" data-more-link-behavior="true" href="#">more</a></div><div class="complete hidden">In this article we describe some elements from the chapter of Mathematics " Number Theory " , as they appear in the Codex Vindobonensis phil. Gr. 65, a Byzantine Ms kept in the National Library of Austria in Vienna. This codex contains a comprehensive program of teaching Mathematics which was addressed to an audience consisting of students probably of all the grades of what is today's primary and secondary education, but also state functionaries, merchants, craftsmen of various specialities such as silversmiths and goldsmiths, builders, etc. The mathematical branch of Number Theory and the numerical position system is integrated in codex 65 in the respective chapters of the four operations and their checks. Introduction At the beginning of this article we briefly describe Codex Vindobonensis phil. Gr. 65 fols 11r ―126r, which was written circa AD 1436 by an anonymous author, and was intended for teaching purposes. We record the syllabus units so that the reader will form an initial idea about the content of the Ms, and then we analyze in detail the problems related to the Number Theory, but also more generally to the numerical system of position in Byzantium.</div></div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/31106546" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="af775ddbe249a713e0c841640ae687ce" rel="nofollow" data-download="{"attachment_id":51541833,"asset_id":31106546,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/51541833/download_file?st=MTc0MDg2MzQyOCw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="50492709" href="https://en-uoa-gr.academia.edu/DrChalkou">Maria Chalkou, State High School Advisor, PhD with "Summa cum Laude" (NKUA), MSc mathematician, editor of Codex Vindobonensis phil. Gr. 65, 15th cent. ff.(11r-126r)-2006, and Codex 72 of the Library of Dimitsana, 18th cent.-2009</a><script data-card-contents-for-user="50492709" type="text/json">{"id":50492709,"first_name":"Maria Chalkou, State High School Advisor, PhD with \"Summa cum Laude\" (NKUA), MSc mathematician, editor of Codex Vindobonensis phil. Gr. 65, 15th cent.","last_name":"ff.(11r-126r)-2006, and Codex 72 of the Library of Dimitsana, 18th cent.-2009","domain_name":"en-uoa-gr","page_name":"DrChalkou","display_name":"Maria Chalkou, State High School Advisor, PhD with \"Summa cum Laude\" (NKUA), MSc mathematician, editor of Codex Vindobonensis phil. 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class="summarized">Finite differences of values of the Riemann zeta function at the integers are explored. Such quantities, which occur as coefficients in Newton series representations, have surfaced in works of Bombieri-Lagarias, Maślanka, Coffey,... <a class="more_link u-tcGrayDark u-linkUnstyled" data-container=".work_3221292" data-show=".complete" data-hide=".summarized" data-more-link-behavior="true" href="#">more</a></div><div class="complete hidden">Finite differences of values of the Riemann zeta function at the integers are explored. Such quantities, which occur as coefficients in Newton series representations, have surfaced in works of Bombieri-Lagarias, Maślanka, Coffey, Báez-Duarte, Voros and others. We apply the theory of Nörlund-Rice integrals in conjunction with the saddlepoint method and derive precise asymptotic estimates. The method extends to Dirichlet Lfunctions and our estimates appear to be partly related to earlier investigations surrounding Li's criterion for the Riemann hypothesis. √ πn .</div></div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/3221292" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="774e8276f1b4460181184569e76bcc68" rel="nofollow" data-download="{"attachment_id":50407190,"asset_id":3221292,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/50407190/download_file?st=MTc0MDg2MzQyOCw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="3685531" href="https://independent.academia.edu/LinasVepstas">Linas Vepstas</a><script data-card-contents-for-user="3685531" type="text/json">{"id":3685531,"first_name":"Linas","last_name":"Vepstas","domain_name":"independent","page_name":"LinasVepstas","display_name":"Linas Vepstas","profile_url":"https://independent.academia.edu/LinasVepstas?f_ri=302","photo":"https://gravatar.com/avatar/a221b4b4e6dd199750731c51ee864a91?s=65"}</script></span></span></li><li class="js-paper-rank-work_3221292 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="3221292"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 3221292, container: ".js-paper-rank-work_3221292", }); 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Such quantities, which occur as coefficients in Newton series representations, have surfaced in works of Bombieri-Lagarias, Maślanka, Coffey, Báez-Duarte, Voros and others. We apply the theory of Nörlund-Rice integrals in conjunction with the saddlepoint method and derive precise asymptotic estimates. The method extends to Dirichlet Lfunctions and our estimates appear to be partly related to earlier investigations surrounding Li's criterion for the Riemann hypothesis. √ πn .","downloadable_attachments":[{"id":50407190,"asset_id":3221292,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":3685531,"first_name":"Linas","last_name":"Vepstas","domain_name":"independent","page_name":"LinasVepstas","display_name":"Linas Vepstas","profile_url":"https://independent.academia.edu/LinasVepstas?f_ri=302","photo":"https://gravatar.com/avatar/a221b4b4e6dd199750731c51ee864a91?s=65"}],"research_interests":[{"id":302,"name":"Analytic Number Theory","url":"https://www.academia.edu/Documents/in/Analytic_Number_Theory?f_ri=302","nofollow":true},{"id":136936,"name":"Continued Fractions","url":"https://www.academia.edu/Documents/in/Continued_Fractions?f_ri=302","nofollow":true},{"id":229028,"name":"riemann Hypothesis","url":"https://www.academia.edu/Documents/in/riemann_Hypothesis?f_ri=302","nofollow":true},{"id":615173,"name":"Reimann Zeta Function","url":"https://www.academia.edu/Documents/in/Reimann_Zeta_Function?f_ri=302","nofollow":true}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_13084199" data-work_id="13084199" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/13084199/The_Non_Trivial_Zeros_Of_The_Riemann_Zeta_Function">The Non-Trivial Zeros Of The Riemann Zeta Function</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest"><div class="summarized">This paper shows why the non-trivial zeros of the Riemann zeta function ζ will always be on the critical line Re(s) = 1/2 and not anywhere else on the critical strip bounded by Re(s) = 0 and Re(s) = 1, thus affirming the validity of the... <a class="more_link u-tcGrayDark u-linkUnstyled" data-container=".work_13084199" data-show=".complete" data-hide=".summarized" data-more-link-behavior="true" href="#">more</a></div><div class="complete hidden">This paper shows why the non-trivial zeros of the Riemann zeta function ζ will always be on the critical line Re(s) = 1/2 and not anywhere else on the critical strip bounded by Re(s) = 0 and Re(s) = 1, thus affirming the validity of the Riemann hypothesis. [The paper is published in a journal of number theory.]</div></div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/13084199" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="51aa738e1350ef8b5f0ffaeab4595a86" rel="nofollow" data-download="{"attachment_id":59500718,"asset_id":13084199,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/59500718/download_file?st=MTc0MDg2MzQyOCw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="11414130" href="https://independent.academia.edu/WongBertrand">Bertrand Wong</a><script data-card-contents-for-user="11414130" type="text/json">{"id":11414130,"first_name":"Bertrand","last_name":"Wong","domain_name":"independent","page_name":"WongBertrand","display_name":"Bertrand Wong","profile_url":"https://independent.academia.edu/WongBertrand?f_ri=302","photo":"https://0.academia-photos.com/11414130/3329015/3917133/s65_bertrand.wong.jpg"}</script></span></span></li><li class="js-paper-rank-work_13084199 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="13084199"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 13084199, container: ".js-paper-rank-work_13084199", }); 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[The paper is published in a journal of number theory.]","downloadable_attachments":[{"id":59500718,"asset_id":13084199,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":11414130,"first_name":"Bertrand","last_name":"Wong","domain_name":"independent","page_name":"WongBertrand","display_name":"Bertrand Wong","profile_url":"https://independent.academia.edu/WongBertrand?f_ri=302","photo":"https://0.academia-photos.com/11414130/3329015/3917133/s65_bertrand.wong.jpg"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics?f_ri=302","nofollow":true},{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=302","nofollow":true},{"id":302,"name":"Analytic Number Theory","url":"https://www.academia.edu/Documents/in/Analytic_Number_Theory?f_ri=302","nofollow":true},{"id":303,"name":"Algebraic Number 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Proofs","url":"https://www.academia.edu/Documents/in/Mathematical_Reasoning_and_Proofs?f_ri=302"},{"id":1031338,"name":"Analitikal dan Logikal (Analytical and Logical)","url":"https://www.academia.edu/Documents/in/Analitikal_dan_Logikal_Analytical_and_Logical_?f_ri=302"},{"id":1255945,"name":"Solvability","url":"https://www.academia.edu/Documents/in/Solvability?f_ri=302"},{"id":1265874,"name":"Proofs and Mathematical Reasoning","url":"https://www.academia.edu/Documents/in/Proofs_and_Mathematical_Reasoning?f_ri=302"},{"id":1971207,"name":"Mathematical Reasoning","url":"https://www.academia.edu/Documents/in/Mathematical_Reasoning?f_ri=302"}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_49513441" data-work_id="49513441" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/49513441/New_Set_and_Number_Theory">New Set and Number Theory</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest">OBSOLETE version. This initial document has been divided into a series of papers for practical reasons and updated on many occasions. Have thus a look at my drafts.</div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/49513441" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="6582be027ab36c04bedf2e063126254b" rel="nofollow" data-download="{"attachment_id":67848735,"asset_id":49513441,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/67848735/download_file?st=MTc0MDg2MzQyOCw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="197874747" href="https://independent.academia.edu/PetraBanski">Petra Banski</a><script data-card-contents-for-user="197874747" type="text/json">{"id":197874747,"first_name":"Petra","last_name":"Banski","domain_name":"independent","page_name":"PetraBanski","display_name":"Petra Banski","profile_url":"https://independent.academia.edu/PetraBanski?f_ri=302","photo":"https://0.academia-photos.com/197874747/59902098/52157643/s65_petra.banski.png"}</script></span></span></li><li class="js-paper-rank-work_49513441 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="49513441"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 49513441, container: ".js-paper-rank-work_49513441", }); 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This initial document has been divided into a series of papers for practical reasons and updated on many occasions. Have thus a look at my drafts.","downloadable_attachments":[{"id":67848735,"asset_id":49513441,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":197874747,"first_name":"Petra","last_name":"Banski","domain_name":"independent","page_name":"PetraBanski","display_name":"Petra Banski","profile_url":"https://independent.academia.edu/PetraBanski?f_ri=302","photo":"https://0.academia-photos.com/197874747/59902098/52157643/s65_petra.banski.png"}],"research_interests":[{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=302","nofollow":true},{"id":302,"name":"Analytic Number Theory","url":"https://www.academia.edu/Documents/in/Analytic_Number_Theory?f_ri=302","nofollow":true},{"id":303,"name":"Algebraic Number Theory","url":"https://www.academia.edu/Documents/in/Algebraic_Number_Theory?f_ri=302","nofollow":true},{"id":363,"name":"Set Theory","url":"https://www.academia.edu/Documents/in/Set_Theory?f_ri=302","nofollow":true},{"id":2392,"name":"Proof Theory","url":"https://www.academia.edu/Documents/in/Proof_Theory?f_ri=302"},{"id":12474,"name":"Algebraic Combinatorics","url":"https://www.academia.edu/Documents/in/Algebraic_Combinatorics?f_ri=302"},{"id":44397,"name":"Elliptic curves","url":"https://www.academia.edu/Documents/in/Elliptic_curves?f_ri=302"},{"id":72443,"name":"Prime Numbers","url":"https://www.academia.edu/Documents/in/Prime_Numbers?f_ri=302"},{"id":200014,"name":"Goldbach's Conjecture","url":"https://www.academia.edu/Documents/in/Goldbachs_Conjecture?f_ri=302"},{"id":229028,"name":"riemann Hypothesis","url":"https://www.academia.edu/Documents/in/riemann_Hypothesis?f_ri=302"},{"id":256321,"name":"Prime Number Theory","url":"https://www.academia.edu/Documents/in/Prime_Number_Theory?f_ri=302"},{"id":686306,"name":"Pure Mathmatics","url":"https://www.academia.edu/Documents/in/Pure_Mathmatics?f_ri=302"}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_14148249" data-work_id="14148249" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/14148249/Zeta_Functions">Zeta Functions</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest">Riemann Zeta Function,<br />Hurwitz Zeta Function,<br />Epstein Zeta Function,<br />Mellin Transform</div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/14148249" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="428dfc640884aa6e50ef81c9e902a3dd" rel="nofollow" data-download="{"attachment_id":38216387,"asset_id":14148249,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/38216387/download_file?st=MTc0MDg2MzQyOCw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="2792810" href="https://independentresearcher.academia.edu/DrJMAshfaqueAMIMAMInstP">Dr. J. 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M.","last_name":"Ashfaque (MInstP)","domain_name":"independentresearcher","page_name":"DrJMAshfaqueAMIMAMInstP","display_name":"Dr. J. M. Ashfaque (MInstP)","profile_url":"https://independentresearcher.academia.edu/DrJMAshfaqueAMIMAMInstP?f_ri=302","photo":"https://0.academia-photos.com/2792810/914293/18370870/s65_dr._j._m..ashfaque_amima_minstp_.jpg"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics?f_ri=302","nofollow":true},{"id":302,"name":"Analytic Number Theory","url":"https://www.academia.edu/Documents/in/Analytic_Number_Theory?f_ri=302","nofollow":true},{"id":318,"name":"Mathematical Physics","url":"https://www.academia.edu/Documents/in/Mathematical_Physics?f_ri=302","nofollow":true},{"id":503,"name":"Theoretical Physics","url":"https://www.academia.edu/Documents/in/Theoretical_Physics?f_ri=302","nofollow":true}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_49404499" data-work_id="49404499" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/49404499/Th%C3%A9or%C3%A8me_de_Jordan_Friable">Théorème de Jordan Friable</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest"><div class="summarized">Extending a previous result, we show that, for the friable summation method, the Fourier series of any normalized function F with bounded variation on the unidimensional torus converges pointwise to F while avoiding the Gibbs phenomenon.... <a class="more_link u-tcGrayDark u-linkUnstyled" data-container=".work_49404499" data-show=".complete" data-hide=".summarized" data-more-link-behavior="true" href="#">more</a></div><div class="complete hidden">Extending a previous result, we show that, for the friable summation method, the Fourier series of any normalized function F with bounded variation on the unidimensional torus converges pointwise to F while avoiding the Gibbs phenomenon. We also prove that the convergence is uniform when F is continuous and provide an e↵ective bound for the speed when F satisfies a uniform Lipschitz condition.</div></div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/49404499" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="91771229605b47ae8a63363eb18db20e" rel="nofollow" data-download="{"attachment_id":67765902,"asset_id":49404499,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/67765902/download_file?st=MTc0MDg2MzQyOCw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="59184542" href="https://independent.academia.edu/TenenbaumG%C3%A9rald">Gérald Tenenbaum</a><script data-card-contents-for-user="59184542" type="text/json">{"id":59184542,"first_name":"Gérald","last_name":"Tenenbaum","domain_name":"independent","page_name":"TenenbaumGérald","display_name":"Gérald Tenenbaum","profile_url":"https://independent.academia.edu/TenenbaumG%C3%A9rald?f_ri=302","photo":"https://0.academia-photos.com/59184542/23734469/22761602/s65_g_rald.tenenbaum.jpg"}</script></span></span></li><li class="js-paper-rank-work_49404499 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="49404499"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 49404499, container: ".js-paper-rank-work_49404499", }); 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$(".js-view-count[data-work-id=49404499]").text(description); $(".js-view-count-work_49404499").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_49404499").removeClass('hidden') })</script></div></li><li class="InlineList-item u-positionRelative" style="max-width: 250px"><div class="u-positionAbsolute" data-has-card-for-ri-list="49404499"><i class="fa fa-tag InlineList-item-icon u-positionRelative"></i></div><span class="InlineList-item-text u-textTruncate u-pl6x"><a class="InlineList-item-text" data-has-card-for-ri="302" rel="nofollow" href="https://www.academia.edu/Documents/in/Analytic_Number_Theory">Analytic Number Theory</a><script data-card-contents-for-ri="302" type="text/json">{"id":302,"name":"Analytic Number Theory","url":"https://www.academia.edu/Documents/in/Analytic_Number_Theory?f_ri=302","nofollow":true}</script></span></li><script>(function(){ if (false) { new Aedu.ResearchInterestListCard({ el: $('*[data-has-card-for-ri-list=49404499]'), work: {"id":49404499,"title":"Théorème de Jordan Friable","created_at":"2021-06-26T11:33:11.709-07:00","url":"https://www.academia.edu/49404499/Th%C3%A9or%C3%A8me_de_Jordan_Friable?f_ri=302","dom_id":"work_49404499","summary":"Extending a previous result, we show that, for the friable summation method, the Fourier series of any normalized function F with bounded variation on the unidimensional torus converges pointwise to F while avoiding the Gibbs phenomenon. We also prove that the convergence is uniform when F is continuous and provide an e↵ective bound for the speed when F satisfies a uniform Lipschitz condition.","downloadable_attachments":[{"id":67765902,"asset_id":49404499,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":59184542,"first_name":"Gérald","last_name":"Tenenbaum","domain_name":"independent","page_name":"TenenbaumGérald","display_name":"Gérald Tenenbaum","profile_url":"https://independent.academia.edu/TenenbaumG%C3%A9rald?f_ri=302","photo":"https://0.academia-photos.com/59184542/23734469/22761602/s65_g_rald.tenenbaum.jpg"}],"research_interests":[{"id":302,"name":"Analytic Number Theory","url":"https://www.academia.edu/Documents/in/Analytic_Number_Theory?f_ri=302","nofollow":true}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_74701488" data-work_id="74701488" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/74701488/Fibonacci_and_Fibonacci_type_sequences_In_search_of_innovative_pattern_recognition">Fibonacci and Fibonacci-type sequences: In search of innovative pattern recognition</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest"><div class="summarized">The objective of this study is to explore an innovative pattern along with derivations connected to the Fibonacci sequences. Relevant proofs with supporting explanations have been provided in this study to derive the conceptualized... <a class="more_link u-tcGrayDark u-linkUnstyled" data-container=".work_74701488" data-show=".complete" data-hide=".summarized" data-more-link-behavior="true" href="#">more</a></div><div class="complete hidden">The objective of this study is to explore an innovative pattern along with derivations connected to the Fibonacci sequences. Relevant proofs with supporting explanations have been provided in this study to derive the conceptualized pattern for this study which in turn will ensure reduction of the complexions and simplification of the Fibonacci formula while solving different problems linked to Fibonacci series and Fibonacci-type sequences. An attempt has also been made in this study to validate the conceptualized pattern with the help of induction method.</div></div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/74701488" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="23bdfd59f414f12e69913309f7f13acf" rel="nofollow" data-download="{"attachment_id":82757512,"asset_id":74701488,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/82757512/download_file?st=MTc0MDg2MzQyOCw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="213114644" href="https://independent.academia.edu/ARCHISMANNANDY">ARCHISMAN NANDY</a><script data-card-contents-for-user="213114644" type="text/json">{"id":213114644,"first_name":"ARCHISMAN","last_name":"NANDY","domain_name":"independent","page_name":"ARCHISMANNANDY","display_name":"ARCHISMAN NANDY","profile_url":"https://independent.academia.edu/ARCHISMANNANDY?f_ri=302","photo":"https://0.academia-photos.com/213114644/72076160/60533592/s65_archisman.nandy.jpeg"}</script></span></span></li><li class="js-paper-rank-work_74701488 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="74701488"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 74701488, container: ".js-paper-rank-work_74701488", }); 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$(".js-view-count[data-work-id=74701488]").text(description); $(".js-view-count-work_74701488").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_74701488").removeClass('hidden') })</script></div></li><li class="InlineList-item u-positionRelative" style="max-width: 250px"><div class="u-positionAbsolute" data-has-card-for-ri-list="74701488"><i class="fa fa-tag InlineList-item-icon u-positionRelative"></i>  <a class="InlineList-item-text u-positionRelative">18</a>  </div><span class="InlineList-item-text u-textTruncate u-pl10x"><a class="InlineList-item-text" data-has-card-for-ri="302" rel="nofollow" href="https://www.academia.edu/Documents/in/Analytic_Number_Theory">Analytic Number Theory</a>, <script data-card-contents-for-ri="302" type="text/json">{"id":302,"name":"Analytic Number Theory","url":"https://www.academia.edu/Documents/in/Analytic_Number_Theory?f_ri=302","nofollow":true}</script><a class="InlineList-item-text" data-has-card-for-ri="303" rel="nofollow" href="https://www.academia.edu/Documents/in/Algebraic_Number_Theory">Algebraic Number Theory</a>, <script data-card-contents-for-ri="303" type="text/json">{"id":303,"name":"Algebraic Number Theory","url":"https://www.academia.edu/Documents/in/Algebraic_Number_Theory?f_ri=302","nofollow":true}</script><a class="InlineList-item-text" data-has-card-for-ri="304" rel="nofollow" href="https://www.academia.edu/Documents/in/Geometric_Number_Theory">Geometric Number Theory</a>, <script data-card-contents-for-ri="304" type="text/json">{"id":304,"name":"Geometric Number Theory","url":"https://www.academia.edu/Documents/in/Geometric_Number_Theory?f_ri=302","nofollow":true}</script><a class="InlineList-item-text" data-has-card-for-ri="476" rel="nofollow" href="https://www.academia.edu/Documents/in/Computational_Number_Theory">Computational Number Theory</a><script data-card-contents-for-ri="476" type="text/json">{"id":476,"name":"Computational Number Theory","url":"https://www.academia.edu/Documents/in/Computational_Number_Theory?f_ri=302","nofollow":true}</script></span></li><script>(function(){ if (true) { new Aedu.ResearchInterestListCard({ el: $('*[data-has-card-for-ri-list=74701488]'), work: {"id":74701488,"title":"Fibonacci and Fibonacci-type sequences: In search of innovative pattern recognition","created_at":"2022-03-27T08:45:04.269-07:00","url":"https://www.academia.edu/74701488/Fibonacci_and_Fibonacci_type_sequences_In_search_of_innovative_pattern_recognition?f_ri=302","dom_id":"work_74701488","summary":"The objective of this study is to explore an innovative pattern along with derivations connected to the Fibonacci sequences. Relevant proofs with supporting explanations have been provided in this study to derive the conceptualized pattern for this study which in turn will ensure reduction of the complexions and simplification of the Fibonacci formula while solving different problems linked to Fibonacci series and Fibonacci-type sequences. An attempt has also been made in this study to validate the conceptualized pattern with the help of induction method. ","downloadable_attachments":[{"id":82757512,"asset_id":74701488,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":213114644,"first_name":"ARCHISMAN","last_name":"NANDY","domain_name":"independent","page_name":"ARCHISMANNANDY","display_name":"ARCHISMAN NANDY","profile_url":"https://independent.academia.edu/ARCHISMANNANDY?f_ri=302","photo":"https://0.academia-photos.com/213114644/72076160/60533592/s65_archisman.nandy.jpeg"}],"research_interests":[{"id":302,"name":"Analytic Number Theory","url":"https://www.academia.edu/Documents/in/Analytic_Number_Theory?f_ri=302","nofollow":true},{"id":303,"name":"Algebraic Number Theory","url":"https://www.academia.edu/Documents/in/Algebraic_Number_Theory?f_ri=302","nofollow":true},{"id":304,"name":"Geometric Number Theory","url":"https://www.academia.edu/Documents/in/Geometric_Number_Theory?f_ri=302","nofollow":true},{"id":476,"name":"Computational Number Theory","url":"https://www.academia.edu/Documents/in/Computational_Number_Theory?f_ri=302","nofollow":true},{"id":20254,"name":"Elementary Number Theory","url":"https://www.academia.edu/Documents/in/Elementary_Number_Theory?f_ri=302"},{"id":77989,"name":"Algebraic computational number theory and cryptography","url":"https://www.academia.edu/Documents/in/Algebraic_computational_number_theory_and_cryptography?f_ri=302"},{"id":120735,"name":"Mathematics(Number Theory)","url":"https://www.academia.edu/Documents/in/Mathematics_Number_Theory_?f_ri=302"},{"id":256321,"name":"Prime Number Theory","url":"https://www.academia.edu/Documents/in/Prime_Number_Theory?f_ri=302"},{"id":256867,"name":"Computer Security, Cryptography, Number theory","url":"https://www.academia.edu/Documents/in/Computer_Security_Cryptography_Number_theory?f_ri=302"},{"id":309422,"name":"Fibonacci numbers","url":"https://www.academia.edu/Documents/in/Fibonacci_numbers?f_ri=302"},{"id":408757,"name":"ALGEBRA COMBINATORIAL NUMBER THEORY","url":"https://www.academia.edu/Documents/in/ALGEBRA_COMBINATORIAL_NUMBER_THEORY?f_ri=302"},{"id":421287,"name":"Theory of Numbers","url":"https://www.academia.edu/Documents/in/Theory_of_Numbers?f_ri=302"},{"id":673215,"name":"Fibonacci Sequences","url":"https://www.academia.edu/Documents/in/Fibonacci_Sequences?f_ri=302"},{"id":886326,"name":"Infinite Series, Number Theory","url":"https://www.academia.edu/Documents/in/Infinite_Series_Number_Theory?f_ri=302"},{"id":914479,"name":"Number Theory Problem Solving","url":"https://www.academia.edu/Documents/in/Number_Theory_Problem_Solving?f_ri=302"},{"id":1005964,"name":"Metric Number Theory","url":"https://www.academia.edu/Documents/in/Metric_Number_Theory?f_ri=302"},{"id":1348861,"name":"Additive Number Theory","url":"https://www.academia.edu/Documents/in/Additive_Number_Theory?f_ri=302"},{"id":2425534,"name":"Numbers Theory","url":"https://www.academia.edu/Documents/in/Numbers_Theory?f_ri=302"}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_27935389" data-work_id="27935389" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/27935389/POWER_OF_TWO_AS_SUMS_OF_THREE_PELL_NUMBERS">POWER OF TWO AS SUMS OF THREE PELL NUMBERS</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest"><div class="summarized">In this paper, we find all the solutions of the Diophantine equation P +P m +P n = 2 a , in nonnegative integer variables (n, m, , a) where P k is the k-th term of the Pell sequence {P n } n≥0 given by P 0 = 0, P 1 = 1 and P n+1 = 2P n +P... <a class="more_link u-tcGrayDark u-linkUnstyled" data-container=".work_27935389" data-show=".complete" data-hide=".summarized" data-more-link-behavior="true" href="#">more</a></div><div class="complete hidden">In this paper, we find all the solutions of the Diophantine equation P +P m +P n = 2 a , in nonnegative integer variables (n, m, , a) where P k is the k-th term of the Pell sequence {P n } n≥0 given by P 0 = 0, P 1 = 1 and P n+1 = 2P n +P n−1 for all n ≥ 1. MSC: 11D45, 11B39; 11A25</div></div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/27935389" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="ead963ffd550b333b7f9cafb4b1e0a46" rel="nofollow" data-download="{"attachment_id":48231818,"asset_id":27935389,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/48231818/download_file?st=MTc0MDg2MzQyOCw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="35595588" href="https://fami4real2000.academia.edu/BernadetteFaye">Bernadette Faye</a><script data-card-contents-for-user="35595588" type="text/json">{"id":35595588,"first_name":"Bernadette","last_name":"Faye","domain_name":"fami4real2000","page_name":"BernadetteFaye","display_name":"Bernadette Faye","profile_url":"https://fami4real2000.academia.edu/BernadetteFaye?f_ri=302","photo":"https://0.academia-photos.com/35595588/10382199/11585345/s65_bernadette.faye.png"}</script></span></span></li><li class="js-paper-rank-work_27935389 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="27935389"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 27935389, container: ".js-paper-rank-work_27935389", }); 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MSC: 11D45, 11B39; 11A25","downloadable_attachments":[{"id":48231818,"asset_id":27935389,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":35595588,"first_name":"Bernadette","last_name":"Faye","domain_name":"fami4real2000","page_name":"BernadetteFaye","display_name":"Bernadette Faye","profile_url":"https://fami4real2000.academia.edu/BernadetteFaye?f_ri=302","photo":"https://0.academia-photos.com/35595588/10382199/11585345/s65_bernadette.faye.png"}],"research_interests":[{"id":302,"name":"Analytic Number Theory","url":"https://www.academia.edu/Documents/in/Analytic_Number_Theory?f_ri=302","nofollow":true},{"id":952345,"name":"Pell Numbers","url":"https://www.academia.edu/Documents/in/Pell_Numbers?f_ri=302","nofollow":true}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_9251980" data-work_id="9251980" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/9251980/An_example_of_a_Number_Theory_Math_Olympiad_Problem">An example of a Number Theory Math Olympiad Problem</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest">How does a Number Theory Math Olympiad problem look like?<br />(An example)</div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/9251980" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="f2e9867d314e4278862ed4b1dc6bb3eb" rel="nofollow" data-download="{"attachment_id":35520395,"asset_id":9251980,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/35520395/download_file?st=MTc0MDg2MzQyOCw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="20118759" href="https://ukm.academia.edu/JonathanRamachandran">Jonathan Ramachandran</a><script data-card-contents-for-user="20118759" type="text/json">{"id":20118759,"first_name":"Jonathan","last_name":"Ramachandran","domain_name":"ukm","page_name":"JonathanRamachandran","display_name":"Jonathan Ramachandran","profile_url":"https://ukm.academia.edu/JonathanRamachandran?f_ri=302","photo":"https://0.academia-photos.com/20118759/5649857/20815939/s65_jonathan.ramachandran.jpg"}</script></span></span></li><li class="js-paper-rank-work_9251980 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="9251980"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 9251980, container: ".js-paper-rank-work_9251980", }); 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On the first observation we learn to span all odd composite numbers on the number line in terms of x and y both ∈ N. This gives us an algebraic form that holds only a prime at the point... <a class="more_link u-tcGrayDark u-linkUnstyled" data-container=".work_79098902" data-show=".complete" data-hide=".summarized" data-more-link-behavior="true" href="#">more</a></div><div class="complete hidden">The paper discusses some advances on prime numbers. On the first observation we learn to span all odd composite numbers on the number line in terms of x and y both ∈ N. This gives us an algebraic form that holds only a prime at the point of no solution and thus, helped us span the primes in negation. In the process we could address specifically those numbers that are separated from an odd composite by a gap k in terms of a union of congruence relations. The number of primes in this set is the difference between the number of primes considered let's say N and the number of the prime gap k occurring till the Nth prime. This union can be further reduced to a series of intersections each of which has a unique solution from the Chinese remainder theorem. This then connects to the prime counting functions for a line to the frequency of the prime gap. We show a relation between the prime gaps of 2 and 4 occurring equally often in N and the prime number theorem in arithmetic progressions.</div></div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/79098902" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="1abf601120cc3f66b1849c5a4877c8b0" rel="nofollow" data-download="{"attachment_id":85936588,"asset_id":79098902,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/85936588/download_file?st=MTc0MDg2MzQyOCw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="81847698" href="https://iitdelhi.academia.edu/Cgupta">Chitradeep Gupta</a><script data-card-contents-for-user="81847698" type="text/json">{"id":81847698,"first_name":"Chitradeep","last_name":"Gupta","domain_name":"iitdelhi","page_name":"Cgupta","display_name":"Chitradeep Gupta","profile_url":"https://iitdelhi.academia.edu/Cgupta?f_ri=302","photo":"https://0.academia-photos.com/81847698/19753282/71286420/s65_chitradeep.gupta.jpeg"}</script></span></span></li><li class="js-paper-rank-work_79098902 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="79098902"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 79098902, container: ".js-paper-rank-work_79098902", }); });</script></li><li class="js-percentile-work_79098902 InlineList-item InlineList-item--bordered hidden u-tcGrayDark"><span class="percentile-widget hidden"><span class="u-mr2x percentile-widget" style="display: none">•</span><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 79098902; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-percentile-work_79098902"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></li><li class="js-view-count-work_79098902 InlineList-item InlineList-item--bordered hidden"><div><span><span class="js-view-count view-count u-mr2x" data-work-id="79098902"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 79098902; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=79098902]").text(description); $(".js-view-count-work_79098902").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_79098902").removeClass('hidden') })</script></div></li><li class="InlineList-item u-positionRelative" style="max-width: 250px"><div class="u-positionAbsolute" data-has-card-for-ri-list="79098902"><i class="fa fa-tag InlineList-item-icon u-positionRelative"></i></div><span class="InlineList-item-text u-textTruncate u-pl6x"><a class="InlineList-item-text" data-has-card-for-ri="302" rel="nofollow" href="https://www.academia.edu/Documents/in/Analytic_Number_Theory">Analytic Number Theory</a><script data-card-contents-for-ri="302" type="text/json">{"id":302,"name":"Analytic Number Theory","url":"https://www.academia.edu/Documents/in/Analytic_Number_Theory?f_ri=302","nofollow":true}</script></span></li><script>(function(){ if (false) { new Aedu.ResearchInterestListCard({ el: $('*[data-has-card-for-ri-list=79098902]'), work: {"id":79098902,"title":"A Natural Language of Primes and Composites in Algebraic Forms","created_at":"2022-05-15T03:54:40.010-07:00","url":"https://www.academia.edu/79098902/A_Natural_Language_of_Primes_and_Composites_in_Algebraic_Forms?f_ri=302","dom_id":"work_79098902","summary":"The paper discusses some advances on prime numbers. On the first observation we learn to span all odd composite numbers on the number line in terms of x and y both ∈ N. This gives us an algebraic form that holds only a prime at the point of no solution and thus, helped us span the primes in negation. In the process we could address specifically those numbers that are separated from an odd composite by a gap k in terms of a union of congruence relations. The number of primes in this set is the difference between the number of primes considered let's say N and the number of the prime gap k occurring till the Nth prime. This union can be further reduced to a series of intersections each of which has a unique solution from the Chinese remainder theorem. This then connects to the prime counting functions for a line to the frequency of the prime gap. We show a relation between the prime gaps of 2 and 4 occurring equally often in N and the prime number theorem in arithmetic progressions.","downloadable_attachments":[{"id":85936588,"asset_id":79098902,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":81847698,"first_name":"Chitradeep","last_name":"Gupta","domain_name":"iitdelhi","page_name":"Cgupta","display_name":"Chitradeep Gupta","profile_url":"https://iitdelhi.academia.edu/Cgupta?f_ri=302","photo":"https://0.academia-photos.com/81847698/19753282/71286420/s65_chitradeep.gupta.jpeg"}],"research_interests":[{"id":302,"name":"Analytic Number Theory","url":"https://www.academia.edu/Documents/in/Analytic_Number_Theory?f_ri=302","nofollow":true}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_8419889" data-work_id="8419889" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/8419889/God_may_not_play_dice_with_the_universe_but_something_strange_is_going_on_with_the_prime_numbers">God may not play dice with the universe, but something strange is going on with the prime numbers. </a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest">25 quotes</div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/8419889" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="e17ade5d14fbe863d8cf9817a823517b" rel="nofollow" data-download="{"attachment_id":34811948,"asset_id":8419889,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/34811948/download_file?st=MTc0MDg2MzQyOCw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="7255313" href="https://mit.academia.edu/ConstantineAdraktasPrimeNumbers">Constantine Adraktas</a><script data-card-contents-for-user="7255313" type="text/json">{"id":7255313,"first_name":"Constantine","last_name":"Adraktas","domain_name":"mit","page_name":"ConstantineAdraktasPrimeNumbers","display_name":"Constantine Adraktas","profile_url":"https://mit.academia.edu/ConstantineAdraktasPrimeNumbers?f_ri=302","photo":"https://0.academia-photos.com/7255313/2841768/3846109/s65_constantine.adraktas.jpg"}</script></span></span></li><li class="js-paper-rank-work_8419889 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="8419889"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 8419889, container: ".js-paper-rank-work_8419889", }); 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$(".js-view-count[data-work-id=8419889]").text(description); $(".js-view-count-work_8419889").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_8419889").removeClass('hidden') })</script></div></li><li class="InlineList-item u-positionRelative" style="max-width: 250px"><div class="u-positionAbsolute" data-has-card-for-ri-list="8419889"><i class="fa fa-tag InlineList-item-icon u-positionRelative"></i>  <a class="InlineList-item-text u-positionRelative">7</a>  </div><span class="InlineList-item-text u-textTruncate u-pl9x"><a class="InlineList-item-text" data-has-card-for-ri="301" rel="nofollow" href="https://www.academia.edu/Documents/in/Number_Theory">Number Theory</a>, <script data-card-contents-for-ri="301" type="text/json">{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=302","nofollow":true}</script><a class="InlineList-item-text" data-has-card-for-ri="302" rel="nofollow" href="https://www.academia.edu/Documents/in/Analytic_Number_Theory">Analytic Number Theory</a>, <script data-card-contents-for-ri="302" type="text/json">{"id":302,"name":"Analytic Number Theory","url":"https://www.academia.edu/Documents/in/Analytic_Number_Theory?f_ri=302","nofollow":true}</script><a class="InlineList-item-text" data-has-card-for-ri="49823" rel="nofollow" href="https://www.academia.edu/Documents/in/Distribution_of_Prime_Numbers">Distribution of Prime Numbers</a>, <script data-card-contents-for-ri="49823" type="text/json">{"id":49823,"name":"Distribution of Prime Numbers","url":"https://www.academia.edu/Documents/in/Distribution_of_Prime_Numbers?f_ri=302","nofollow":true}</script><a class="InlineList-item-text" data-has-card-for-ri="72443" rel="nofollow" href="https://www.academia.edu/Documents/in/Prime_Numbers">Prime Numbers</a><script data-card-contents-for-ri="72443" type="text/json">{"id":72443,"name":"Prime Numbers","url":"https://www.academia.edu/Documents/in/Prime_Numbers?f_ri=302","nofollow":true}</script></span></li><script>(function(){ if (true) { new Aedu.ResearchInterestListCard({ el: $('*[data-has-card-for-ri-list=8419889]'), work: {"id":8419889,"title":"God may not play dice with the universe, but something strange is going on with the prime numbers. ","created_at":"2014-09-21T03:37:01.171-07:00","url":"https://www.academia.edu/8419889/God_may_not_play_dice_with_the_universe_but_something_strange_is_going_on_with_the_prime_numbers?f_ri=302","dom_id":"work_8419889","summary":"25 quotes","downloadable_attachments":[{"id":34811948,"asset_id":8419889,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":7255313,"first_name":"Constantine","last_name":"Adraktas","domain_name":"mit","page_name":"ConstantineAdraktasPrimeNumbers","display_name":"Constantine Adraktas","profile_url":"https://mit.academia.edu/ConstantineAdraktasPrimeNumbers?f_ri=302","photo":"https://0.academia-photos.com/7255313/2841768/3846109/s65_constantine.adraktas.jpg"}],"research_interests":[{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=302","nofollow":true},{"id":302,"name":"Analytic Number Theory","url":"https://www.academia.edu/Documents/in/Analytic_Number_Theory?f_ri=302","nofollow":true},{"id":49823,"name":"Distribution of Prime Numbers","url":"https://www.academia.edu/Documents/in/Distribution_of_Prime_Numbers?f_ri=302","nofollow":true},{"id":72443,"name":"Prime Numbers","url":"https://www.academia.edu/Documents/in/Prime_Numbers?f_ri=302","nofollow":true},{"id":256321,"name":"Prime Number Theory","url":"https://www.academia.edu/Documents/in/Prime_Number_Theory?f_ri=302"},{"id":350509,"name":"Leonhard Euler","url":"https://www.academia.edu/Documents/in/Leonhard_Euler?f_ri=302"},{"id":1212174,"name":"Terence Tao","url":"https://www.academia.edu/Documents/in/Terence_Tao?f_ri=302"}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_285114" data-work_id="285114" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/285114/The_continuing_story_of_zeta">The continuing story of zeta</a></div></div><div class="u-pb4x u-mt3x"></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/285114" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="08839bb8f4e0fcd5b1ef2d96ee01ccb5" rel="nofollow" data-download="{"attachment_id":1311204,"asset_id":285114,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/1311204/download_file?st=MTc0MDg2MzQyOCw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="13301" href="https://eastanglia.academia.edu/ThomasWard">Thomas Ward</a><script data-card-contents-for-user="13301" type="text/json">{"id":13301,"first_name":"Thomas","last_name":"Ward","domain_name":"eastanglia","page_name":"ThomasWard","display_name":"Thomas Ward","profile_url":"https://eastanglia.academia.edu/ThomasWard?f_ri=302","photo":"https://0.academia-photos.com/13301/79989/87828/s65_thomas.ward.jpg"}</script></span></span></li><li class="js-paper-rank-work_285114 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="285114"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 285114, container: ".js-paper-rank-work_285114", }); 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In this paper, we show that there only finitely many positive integers d which are not squares, such that the Pell equation X 2 −dY 2 = 1 has two positive integer solutions (X, Y) with the property that their... <a class="more_link u-tcGrayDark u-linkUnstyled" data-container=".work_25639605" data-show=".complete" data-hide=".summarized" data-more-link-behavior="true" href="#">more</a></div><div class="complete hidden">Let b ≥ 2 be a given integer. In this paper, we show that there only finitely many positive integers d which are not squares, such that the Pell equation X 2 −dY 2 = 1 has two positive integer solutions (X, Y) with the property that their X-coordinates are base b-repdigits. 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In this paper, we show that there only finitely many positive integers d which are not squares, such that the Pell equation X 2 −dY 2 = 1 has two positive integer solutions (X, Y) with the property that their X-coordinates are base b-repdigits. We also give an upper bound on the largest such d in terms of b.","downloadable_attachments":[{"id":54861236,"asset_id":25639605,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":35595588,"first_name":"Bernadette","last_name":"Faye","domain_name":"fami4real2000","page_name":"BernadetteFaye","display_name":"Bernadette Faye","profile_url":"https://fami4real2000.academia.edu/BernadetteFaye?f_ri=302","photo":"https://0.academia-photos.com/35595588/10382199/11585345/s65_bernadette.faye.png"}],"research_interests":[{"id":302,"name":"Analytic Number Theory","url":"https://www.academia.edu/Documents/in/Analytic_Number_Theory?f_ri=302","nofollow":true},{"id":303,"name":"Algebraic Number Theory","url":"https://www.academia.edu/Documents/in/Algebraic_Number_Theory?f_ri=302","nofollow":true}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_14449209" data-work_id="14449209" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/14449209/The_Significance_Of_The_Non_Trivial_Zeros_Of_The_Riemann_Zeta_Function">The Significance Of The Non-Trivial Zeros Of The Riemann Zeta Function</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest">This paper expounds the role of the non-trivial zeros of the Riemann zeta function ζ and supplements the author’s earlier papers on the Riemann hypothesis. There is a lot of mystery surrounding the non-trivial zeros.</div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/14449209" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="ec130047bcbad1710ac369cc1967137c" rel="nofollow" data-download="{"attachment_id":38327855,"asset_id":14449209,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/38327855/download_file?st=MTc0MDg2MzQyOCw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="11414130" href="https://independent.academia.edu/WongBertrand">Bertrand Wong</a><script data-card-contents-for-user="11414130" type="text/json">{"id":11414130,"first_name":"Bertrand","last_name":"Wong","domain_name":"independent","page_name":"WongBertrand","display_name":"Bertrand Wong","profile_url":"https://independent.academia.edu/WongBertrand?f_ri=302","photo":"https://0.academia-photos.com/11414130/3329015/3917133/s65_bertrand.wong.jpg"}</script></span></span></li><li class="js-paper-rank-work_14449209 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="14449209"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 14449209, container: ".js-paper-rank-work_14449209", }); 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