Lectures on the Riemann zeta function

130 Pages·2014·0.573 MB·English

by  Iwaniec H.

University
L
ECTURE
Series
Volume 62
Lectures on the
Riemann Zeta Function
H. Iwaniec
American Mathematical Society
Lectures on the
Riemann Zeta Function

University
L
ECTURE
Series
Volume 62
Lectures on the
Riemann Zeta Function
H. Iwaniec
American Mathematical Society
Providence, Rhode Island
EDITORIAL COMMITTEE
Jordan S. Ellenberg Robert Guralnik
WilliamP. Minicozzi II (Chair) Tatiana Toro
2010 Mathematics Subject Classification. Primary 11N05; Secondary 11N37.
For additional informationand updates on this book, visit
www.ams.org/bookpages/ulect-62
Library of Congress Cataloging-in-Publication Data
Iwaniec,Henryk.
LecturesontheRiemannzetafunction/H.Iwaniec.
pagescm. —(Universitylectureseries;volume62)
Includesbibliographicalreferencesandindex.
ISBN978-1-4704-1851-9(alk.paper)
1. Riemann, Bernhard, 1826–1866. 2. Functions, Zeta. 3. Riemann hypothesis.
4.Numbers,Prime. I.Title. II.Title: Riemannzetafunction.
QA351.I93 2014
515(cid:2).56—dc23
2014021164
Copying and reprinting. Individual readers of this publication, and nonprofit libraries
actingforthem,arepermittedtomakefairuseofthematerial,suchastocopyachapterforuse
in teaching or research. Permission is granted to quote brief passages from this publication in
reviews,providedthecustomaryacknowledgmentofthesourceisgiven.
Republication,systematiccopying,ormultiplereproductionofanymaterialinthispublication
is permitted only under license from the American Mathematical Society. Requests for such
permissionshouldbeaddressedtotheAcquisitionsDepartment,AmericanMathematicalSociety,
201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by
[email protected].
(cid:2)c 2014bytheAmericanMathematicalSociety. Allrightsreserved.
PrintedintheUnitedStatesofAmerica.
(cid:2)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines
establishedtoensurepermanenceanddurability.
VisittheAMShomepageathttp://www.ams.org/
10987654321 191817161514
Contents
Preface vii
Part 1. Classical Topics 1
Chapter 1. Panorama of Arithmetic Functions 3
Chapter 2. The Euler–Maclaurin Formula 9
Chapter 3. Tchebyshev’s Prime Seeds 13
Chapter 4. Elementary Prime Number Theorem 15
Chapter 5. The Riemann Memoir 21
Chapter 6. The Analytic Continuation 23
Chapter 7. The Functional Equation 25
Chapter 8. The Product Formula over the Zeros 27
Chapter 9. The Asymptotic Formula for N(T) 33
Chapter 10. The Asymptotic Formula for ψ(x) 37
Chapter 11. The Zero-free Region and the PNT 41
Chapter 12. Approximate Functional Equations 43
Chapter 13. The Dirichlet Polynomials 47
Chapter 14. Zeros off the Critical Line 55
Chapter 15. Zeros on the Critical Line 57
Part 2. The Critical Zeros after Levinson 63
Chapter 16. Introduction 65
Chapter 17. Detecting Critical Zeros 67
Chapter 18. Conrey’s Construction 69
Chapter 19. The Argument Variations 71
Chapter 20. Attaching a Mollifier 75
v
vi CONTENTS
Chapter 21. The Littlewood Lemma 77
Chapter 22. The Principal Inequality 79
Chapter 23. Positive Proportion of the Critical Zeros 81
Chapter 24. The First Moment of Dirichlet Polynomials 83
Chapter 25. The Second Moment of Dirichlet Polynomials 85
Chapter 26. The Diagonal Terms 87
Chapter 27. The Off-diagonal Terms 95
Chapter 28. Conclusion 103
Chapter 29. Computations and the Optimal Mollifier 107
Appendix A. Smooth Bump Functions 111
Appendix B. The Gamma Function 115
Bibliography 117
Index 119
Preface
TheRiemannzetafunctionζ(s)intherealvariableswasintroducedbyL.Eu-
ler (1737) in connection with questions about the distribution of prime numbers.
LaterB.Riemann(1859)deriveddeeperresultsabouttheprimenumbersbyconsid-
eringthe zetafunction in the complex variable. He revealeda dual correspondence
between the primes and the complex zeros of ζ(s), which started a theory to be
developed by the greatest minds in mathematics. Riemann was able to provide
proofs of his most fundamental observations, except for one, which asserts that all
the non-trivial zeros of ζ(s) are on the line Res= 1. This is the famous Riemann
2
Hypothesis–oneofthemostimportantunsolvedproblemsinmodernmathematics.
These lecture notes cover closely the material which I presented to graduate
students at Rutgers in the fall of 2012. The theory of the Riemann zeta function
has expanded in different directions over the past 150 years; however my goal was
limitedtoshowingonlyafewclassicalresultsonthedistributionofthezeros. These
results include the Riemann memoir (1859), the density theorem of F. Carlson
(1920) about the zeros off the critical line, and the estimates of G. H. Hardy - J.
E. Littlewood (1921) for the number of zeros on the critical line.
Then,inPart2oftheselectures,IpresentinfulldetailtheresultofN.Levinson
(1974), which asserts that more than one third of the zeros are critical (lie on the
line Res = 1). My approach had frequent detours so that students could learn
2
different techniques with interesting features. For instance, I followed the stronger
constructioninventedbyJ.B.Conrey(1983),becauseitrevealsclearlytheesssence
of Levinson’s ideas.
After establishing the principal inequality of the Levinson-Conrey method, it
remainstoevaluateasymptoticallythesecondpower-momentofarelevantDirichlet
polynomial, which is built out of derivatives of the zeta function and its mollifier.
This task was carried out differently than by the traditional arguments and in
greatergeneralitythanitwasneeded. Themaintermcomingfromthecontribution
of the diagonal terms fits with results in sieve theory and can be useful elsewhere.
I am pleased to express my deep appreciation to Pedro Henrique Pontes, who
actively participated in the course and he gave valuable mathematical comments,
whichimprovedmypresentation. Healsohelpedsignificantlyineditingthesenotes
inadditiontotypingthem. MythanksalsogototheEditorsoftheAMSUniversity
LectureSeriesforpublishingthesenotesintheirvolumes,andinparticulartoSergei
Gelfand for continuous encouragements.
vii

Part 1
Classical Topics