Document Zbl 0315.10035

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M.</a></div> <h2 class="title"> <strong>Riemann’s zeta function.</strong> <i>(English)</i> <a class="label nowrap" href="/0315.10035">Zbl 0315.10035</a> </h2> <div class="source"> <a href="/serials/4240" title="Series Profile">Pure and Applied Mathematics, 58. New York-London: Academic Press, a subsidiary of Harcourt Brace Jovanovich, Publishers. XIII, 315 pp. $21.50 (1974).</a> </div> <div class="abstract">Dieses ist ein wichtiges und ungew枚hnliches Buch. Es m枚ge wohl dem mit der Theorie der Primzahlen vertrauten Leser nicht viele neue Tatsachen bringen, es ist jedoch Tr盲ger einer Botschaft: man gehe an die Quellen, man komme stets auf die Meister zur眉ck!<br class="zbmathjax-paragraph">In den meisten F盲chern ist das ja selbstverst盲ndlich. Welcher Literaturforscher w眉rde sich denn damit begn眉gen nur 眉ber Goethe und Schiller zu lesen, ohne sich in die Werke dieser Dichter selber zu vertiefen? In der Mathematik scheint es leider anders zu sein. Sogar solche bahnbrechenden Arbeiten wie “Fundamenta Nova”, “Disquisitiones Arithmeticae”, oder “脺ber die Anzahl der Primzahlen unter einer gegebenen Gr枚脽e” werden heute kaum mehr gelesen, von so manchen anderen wichtigen, jedoch weniger ber眉hmten Werken gar nicht mehr zu reden. Zum Teil m枚ge das auf sprachliche Schwierigkeiten zur眉ckzuf眉hren sein, obwohl so manche lateinische, russische und franz枚siche Texte heute sowohl in deutscher als auch in englischer Fassung zu finden sind. Eine Ausnahme bildete bis jetzt die letzterw盲hnte Arbeit Riemanns. Als Anhang zum gegenw盲rtigen Buch befindet sich jedoch eine dem Verf. zu verdankende englische 脺bersetzung dieses kurzen, jedoch bedeutenden Werkes. Dadurch wird dieses auch denjenigen Mathematikern zug盲nglich gemacht, die der deutschen Sprache nicht gen眉gend m盲chtig sind, um Riemann im Original zu lesen.<br class="zbmathjax-paragraph">Es mag 眉berhaupt wohl zutreffen, da脽 es haupts盲chlich die gegenw盲rtige, relative Vernachl盲ssigung des Riemannschen Beitrages zum Problem der Primzahlen war, die den Verf. veranla脽te das Buch zu schreiben; dieses ungeachtet der Tatsache, da脽<span class="zbmathjax-space"></span>nur etwa 100 (von 眉ber 300) Seiten unmittelbare Beziehung zu Riemanns Arbeiten haben (Kapitel 1, 2, 3 und 7). Jedoch sind auch die anderen 2/3 des Buches von Riemannschen Ideen durchdrungen. Von besonderem Interesse ist die gro脽e Sorgf盲ltigkeit, mit der alle Einzelheiten ausgearbeitet sind, unter Vermeidung jeder Unklarheit, um diesen an sich schwierigen Gegenstand auch dem weniger gewandten Leser zug盲nglich zu machen.<br class="zbmathjax-paragraph">Nur die k眉rzeste 脺bersicht des behandelten Stoffes ist hier m枚glich. Das erste Kapitel ist eine ausf眉hrliche Darlegung von Riemanns “脺ber die Anzahl …”.<br class="zbmathjax-paragraph">Im zweiten Kapitel werden L眉cken jener Arbeit ausgef眉llt (Hadamard, Jensen, Landau,<br class="zbmathjax-newline">Carath茅odory). <br class="zbmathjax-paragraph">Kapitel 3: Beweis der von Mangoldtschen (“expliziten”) Formel f眉r $\psi(x)$ und der entsprechenden Riemannschen Formel f眉r $\pi(x)$ (von Mangoldt, Landau). <br class="zbmathjax-paragraph">Kapitel 4: Beweis des Primzahlsatzes (Hadamard, Chebyshev, Tauberscher Satz). <br class="zbmathjax-paragraph">Kapitel 5: Primzahlsatz mit Restglied. Bedeutung der Riemannschen Vermutung (im folgenden mit RV abgek眉rzt) (Mertens, de la Vall茅e-Poussin, von Koch). <br class="zbmathjax-paragraph">Kapitel 6: Euler-Maclaurinsche Summenformel, Stirlingsche Reihe, Berechnung der Wurzeln von $\zeta(s)$ (Gram, B盲cklund, Hutchinson, Stieltjes). <br class="zbmathjax-paragraph">Kapitel 7: Riemann-Siegelsche Formel, Besprechung der RV (Riemann, Siegel, Bessel-Hagen). <br class="zbmathjax-paragraph">Kapitel 8: Berechnung der Wurzeln im Gro脽en (Turing, Lehmer, Rosser, Yoh茅, Schoenfeld). <br class="zbmathjax-paragraph">Kapitel 9: Beziehung zwischen der Gr枚脽enordung von $|\zeta(s)|$ und der Verteilung der Nullstellen, Lindel枚fsche Vermutung, Mittelwerte, Bohr-Landauscher Satz, bester zur Zeit bekannte Wert des Restgliedes im Primzahlsatz (Hadamard, Littlewood, Lindel枚f, B盲cklund, Bohr, Landau). <br class="zbmathjax-paragraph">Das zehnte Kapitel 眉ber Fouriersche Analyse bespricht invariante Operatoren, Umkehrungss盲tze, usw., und wendet diese Begriffe auf die Zetafunktion an. Auf diese Weise erh盲lt man, unter anderem eine operatorentechnische Variante des zweiten Riemannschen Beweises der Funktionalgleichung der Zetafunktion. Auch Parsevals Gleichung, M枚biussche Umkehrung, Werte von $\zeta(s)$ f眉r ganzrationales $s$ und eine Formel von Ramanujan kommen hier vor. Trotz seiner formellen Eleganz scheint jedoch dieses Kapitel das am wenigsten befriedigende dieses sonst hervorrangenden Buches zu sein. <br class="zbmathjax-paragraph">Kapitel 11 ist den Nullstellen von $\zeta(s)$ auf der kritischen Geraden gewidmet. Die Beweise von Hardy (“unendlich viele Wurzeln”), Hardy und Littlewood (“mehr als $CT$”) und Selberg (“mehr als $CT\log T$”) werden sehr sorgf盲ltig entwickelt. Levinsons neues (1974) Ergebnis (“$C>1/6\pi$”) wird in einer zweiten Auflage erw盲hnt. <br class="zbmathjax-paragraph">Das zw枚lfte und letzte Kapitel behandelt mehrere unzusammenh盲ngende Themen, unter denen folgende erw盲hnt werden m枚gen: Die RV und die Gr枚脽enordnung der M枚biusschen Summenfunktion (Mertens, Stieltjes); Fareysche Reihen und die RV (Franel, Landau); wahrscheinlichkeitstheoretische Betrachtung der RV (Denjoy); Taubersche Lehrs盲tze (Tauber, Hardy, Littlewood, Wiener, Ikehara); Selbergs Ungleichung und der elementare Beweis des Primzahlsatzes (Selberg, Erd艖s, Wirsing, Bombieri); Zahlentheorie und von der Riemannschen verschiedene Zetafunktionen (Dedekind, Hilbert, Hecke, Artin, Weil, Tate). Wie bereits erw盲hnt, folgt ein Anhang mit der englischen 脺bersetzung von Riemanns “脺ber die Anzahl…”, der das Buch beschlie脽t.<br class="zbmathjax-paragraph">Hier angekommen, dr盲ngt sich wohl dem Leser die Frage auf: Was ist des Verf. eigene Meinung bez眉glich der RV? Der Verf. gibt sich ernste M眉he zu zeigen, da脽<span class="zbmathjax-space"></span>nichts von dem, was wir heute 眉ber die Riemannschen Zetafunktionen wissen, uns berechtigt anzunehmen, da脽<span class="zbmathjax-space"></span>die RV zutrifft. Er geht sogar so weit anzunehmen, da脽<span class="zbmathjax-space"></span>wohl Riemann selber, falls er z.B. die Ausnahmen zum Gramschen Prinzip, oder die “Lehmerschen Erscheinung” gekannt h盲tte, wom枚glich seine eigene Vermutung angezweifelt h盲tte. Andererseits werden s盲mtliche Erw盲gungen die den Glauben an die RV bekr盲tigen k枚nnen, sorf盲ltig dem Leser vorgelegt. Am Ende erw盲hnt der Verf. (in einem viel zu kurzen, letzten Abschnitt) die Zetafunktionen der Funktionenk枚rper 眉ber endlichen K枚rpern und die Tatsache, da脽<span class="zbmathjax-space"></span>A. Weil f眉r diese die G眉ltigkeit der analogen “Vermutung” bewiesen hat. Verf. betrachtet diese und verwandte Entwicklungen als die 眉berzeugendsten Argumente zu Gunsten der RV. Obwohl Verf. also keinerlei entg眉ltige Meinung 盲u脽ert, so versteht der Leser doch dessen Empfinden, da脽“…vielleicht doch…”.<br class="zbmathjax-paragraph">Endlich sei der Leser noch einmal an die Botschaft erinnert: kein Kommentar kann Urquellen ersetzen. Im vorliegenden Falle gilt das (abgesehen von Riemanns Werk) ganz besonders von Selbergs Arbeiten. Das gegenw盲rtige Buch mag jedoch das Lesen dieser, so wie mancher anderer Originalarbeiten erheblich erleichtern.<div class="reviewer"> Reviewer: <a href="/authors/?q=E%2A+Grosswald">E. Grosswald</a></div> <div class="clearfix"></div><div class="scan"> <span> Page: <a class="goto" href="/scans/315/062.gif">−5</a> <a class="goto" href="/scans/315/063.gif">−4</a> <a class="goto" href="/scans/315/064.gif">−3</a> <a class="goto" href="/scans/315/065.gif">−2</a> <a class="goto" href="/scans/315/066.gif">−1</a> <a class="goto" href="/scans/315/067.gif">±0</a> <a class="goto" href="/scans/315/068.gif">+1</a> <a class="goto" href="/scans/315/069.gif">+2</a> <a class="goto" href="/scans/315/070.gif">+3</a> <a class="goto" href="/scans/315/071.gif">+4</a> <a class="goto" href="/scans/315/072.gif">+5</a></span> <a class="scan" href="/scans/315/067.gif">Show Scanned Page</a><br> <img src="/scans/315/067.gif" alt="scan of Zbl 0315.10035" onclick="$(this).siblings('.scan').click();"> </div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A3492781">Cited in <strong>6</strong> Reviews</a></div><div class="clear"><a href="/?q=rf%3A3492781">Cited in <strong>266</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A11M06" title="MSC2020">11M06</a> </td> <td class="space"> $\zeta (s)$ and $L(s, \chi)$ </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A11-02" title="MSC2020">11-02</a> </td> <td class="space"> Research exposition (monographs, survey articles) pertaining to number theory </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A11-03" title="MSC2020">11-03</a> </td> <td class="space"> History of number theory </td> </tr></table> </div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 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data-itemname="Zbl 0315.10035" data-ciurl="" data-biburl="/bibtex/03492781.bib" data-amsurl="/amsrefs/03492781.bib" data-xmlurl="/xml/03492781.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/03492781.pdf" title="Zbl 0315.10035 as PDF">Review PDF</a> </div> <div class="fulltexts"> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="mathoverflow"> <h3>Digital Library of Mathematical Functions:</h3> <a href="https://dlmf.nist.gov/25#info">Chapter 25 Zeta and Related Functions</a><br> </div> <div class="mathoverflow"> <h3>Online Encyclopedia of Integer Sequences:</h3> <a href="https://oeis.org/A2410">Nearest integer to imaginary part of n-th zero of Riemann zeta function.</a><br> <a href="https://oeis.org/A13629">Floor of imaginary parts of nontrivial zeros of Riemann zeta function.</a><br> <a href="https://oeis.org/A27641">Numerator of Bernoulli number B_n.</a><br> <a href="https://oeis.org/A74760">Decimal expansion of lambda(1) in Li’s criterion.</a><br> <a href="https://oeis.org/A99319">Numerators of an approximation of Riemann to pi(n).</a><br> <a href="https://oeis.org/A99320">Denominators of an approximation of Riemann to pi(n).</a><br> <a href="https://oeis.org/A117536">Nearest integer to locations of increasingly large peaks of abs(zeta(0.5 + i*2*(Pi/log(2))*t)) for increasing real t.</a><br> <a href="https://oeis.org/A240961">Decimal expansion of y_126, the [imaginary part of the] first zero of Riemann’s zeta function where Gram’s law fails.</a><br> </div> </article> </div></div> </div> </div> <div class="clearfix"></div> </div> </div> <div id="foot"><div class="copyright"> © 2025 <a target="fiz" href="https://www.fiz-karlsruhe.de/en">FIZ Karlsruhe GmbH</a> <a href="/privacy-policy/">Privacy Policy</a> <a href="/legal-notices/">Legal Notices</a> <a href="/terms-conditions/">Terms & Conditions</a> <div class="info"> <ul class="nav"> <li class="mastodon"> <a 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