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STUDENT MATHEMATICAL LIBRARY Volume 92 The Great Prime Number Race Roger Plymen The Great Prime Number Race STUDENT MATHEMATICAL LIBRARY Volume 92 The Great Prime Number Race Roger Plymen EDITORIAL COMMITTEE John McCleary Kavita Ramanan Rosa C. Orellana John Stillwell(Chair) Cover image courtesy of Brandon Carter. 2020 Mathematics Subject Classification. Primary 11-03, 11A41, 11M06, 11N05. For additional informationand updates on this book, visit www.ams.org/bookpages/stml-92 Library of Congress Cataloging-in-Publication Data Names: Plymen,RogerJ.,author. Title: Thegreatprimenumberrace/RogerPlymen. Description: Providence, Rhode Island : American Mathematical Society, 2020. |Series: Student mathematicallibrary, 1520-9121; volume 92 |Includes bib- liographicalreferencesandindex. Identifiers: LCCN2020025241|ISBN9781470462574(paperback)| ISBN9781470462796(ebook) Subjects: LCSH: Numbers, Prime. | Number theory. | AMS: Number theory – Historical|Numbertheory–Elementarynumbertheory–Primes. |Number theory– Zetaand L-functions: analytic theory–ζ(s) andL(s,χ). |Number theory–Multiplicativenumbertheory–Distributionofprimes. Classification: LCCQA246.P6962020|DDC512.7/23–dc23 LCrecordavailableathttps://lccn.loc.gov/2020025241 Copying and reprinting. Individual readers of this publication, and nonprofit li- braries acting for them, are permitted to make fair use of the material, such as to copyselectpagesforuseinteachingorresearch. Permissionisgrantedtoquotebrief passagesfromthispublicationinreviews,providedthecustomaryacknowledgmentof thesourceisgiven. Republication,systematiccopying,ormultiplereproductionofanymaterialinthis publicationispermittedonlyunderlicensefromtheAmericanMathematicalSociety. Requests for permission to reuse portions of AMS publication content are handled by the Copyright Clearance Center. For more information, please visit www.ams.org/ publications/pubpermissions. Send requests for translation rights and licensed reprints to reprint-permission @ams.org. (cid:2)c 2020bytheAmericanMathematicalSociety. Allrightsreserved. TheAmericanMathematicalSocietyretainsallrights exceptthosegrantedtotheUnitedStatesGovernment. PrintedintheUnitedStatesofAmerica. (cid:2)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines establishedtoensurepermanenceanddurability. VisittheAMShomepageathttps://www.ams.org/ 10987654321 252423222120 To Hilary Contents Preface ix Chapter 1. The Riemann zeta function 1 §1.1. Introduction 1 §1.2. The Riemann zeta function 3 §1.3. The prime numbers 4 §1.4. The Riemann zeta function 6 §1.5. Euler and the zeta function 8 §1.6. Meromorphic continuation of ζ(s) 10 Chapter 2. The Euler product 17 §2.1. The zeta function and the Euler product 17 §2.2. The logarithmic derivative of ζ(s) 20 Chapter 3. The functional equation 27 §3.1. The gamma function 29 §3.2. The functional equation 36 §3.3. Some zeta values 41 §3.4. Euler and the functional equation 43 §3.5. The Euler constant revisited 47 vii viii Contents Chapter 4. The explicit formulas in analytic number theory 57 §4.1. The von Mangoldt explicit formula 58 §4.2. Can you hear the Riemann hypothesis? 61 §4.3. Comparison with Fourier series 63 §4.4. Proof of the von Mangoldt formula 65 §4.5. The logarithmic integral Li(z) 69 §4.6. The Riemann formula 73 §4.7. Origin of the Riemann explicit formula 78 Chapter 5. The prime number theorem 81 §5.1. The Riemann-Ramanujan approximation 81 §5.2. Proof of the prime number theorem 82 Chapter 6. Oscillation of π(x)−Li(x) 93 §6.1. Littlewood’s theorem 93 §6.2. Lehman’s theorem 98 Chapter 7. The prime number race 107 §7.1. On the logarithmic density 107 §7.2. Upper bounds for the Skewes number 109 Chapter 8. Exercises, hints, and selected solutions 113 §8.1. Exercises 113 §8.2. Hints and selected solutions 120 Bibliography 133 Index 137 Preface This is a book about prime numbers. Let π(x) denote the number of primes less than or equal to x, and let Li(x) denote the logarithmic integral of x. According to the prime number theorem, we have π(x)∼Li(x). This is an asymptotic statement: it means that π(x) →1 as x→∞. Li(x) If you look at any table of primes, you will find that π(x) < Li(x). Thenumericalevidenceforthisisquitestrong. Infact,weknowthat π(x)<Li(x) for 2≤x≤1019; see [But, Theorem 2(1.10)]. Nevertheless,itremainsafact,provenbyLittlewoodin1914[Li], that the sign of π(x)−Li(x) oscillates infinitely often! It seems appropriate to describe this phenomenon as a “race” between π(x) and Li(x). Sometimes Li(x) is ahead; sometimes π(x) is ahead. According to the numerical evidence, Li(x) starts in front. So, when is the first time that π(x) gets ahead? That is to say, what is the smallest natural number for which the prime number theorem undercounts the number of primes? ix