The Riemann Hypothesis: A Resource for the Afficionado and Virtuoso Alike

527 Pages·2007·31.23 MB·English

by  Peter Borwein

CanadianMathematicalSociety
SociétémathématiqueduCanada
Editors-in-Chief
Rédacteurs-en-chef
K.Dilcher
K.Taylor
AdvisoryBoard
Comitéconsultatif
P.Borwein
R.Kane
S.Shen
CMSBooksinMathematics
OuvragesdemathématiquesdelaSMC
1 Herrman/Kucˇera/Šimsˇa EquationsandInequalities
2 Arnold AbelianGroupsandRepresentationsofFinitePartiallyOrderedSets
3 Borwein/Lewis ConvexAnalysisandNonlinearOptimization,2ndEd.
4 Levin/Lubinsky OrthogonalPolynomialsforExponentialWeights
5 Kane ReflectionGroupsandInvariantTheory
6 Phillips TwoMillenniaofMathematics
7 Deutsch BestApproximationinInnerProductSpaces
8 Fabianetal. FunctionalAnalysisandInfinite-DimensionalGeometry
9 Krˇízˇek/Luca/Somer 17LecturesonFermatNumbers
10 Borwein ComputationalExcursionsinAnalysisandNumberTheory
11 Reed/Sales(Editors) RecentAdvancesinAlgorithmsandCombinatorics
12 Herman/Kucˇera/Šimsˇa CountingandConfigurations
13 Nazareth DifferentiableOptimizationandEquationSolving
14 Phillips InterpolationandApproximationbyPolynomials
15 Ben-Israel/Greville GeneralizedInverses,2ndEd.
16 Zhao DynamicalSystemsinPopulationBiology
17 Göpfertetal. VariationalMethodsinPartiallyOrderedSpaces
18 Akivis/Goldberg DifferentialGeometryofVarietieswithDegenerateGaussMaps
19 Mikhalev/Shpilrain/Yu CombinatorialMethods
20 Borwein/Zhu TechniquesofVariationalAnalysis
21 VanBrummelen/Kinyon MathematicsandtheHistorian’sCraft
22 Lucchetti ConvexityandWell-PosedProblems
23 Niculescu/Persson ConvexFunctionsandTheirApplications
24 Singer DualityforNonconvexApproximationandOptimization
25 Higginson/Pimm/Sinclair MathematicsandtheAesthetic
26 Hájek/Santaluc´ia/Vanderwerff/Zizler BiorthogonalSystemsinBanachSpaces
27 Borwein/Choi/Rooney/Weirathmueller TheRiemannHypothesis
PeterBorwein,StephenChoi,BrendanRooney
andAndreaWeirathmueller(Eds.)
The Riemann Hypothesis
A Resource for the Afficionado
and Virtuoso Alike
PeterBorwein StephenChoi
DepartmentofMathematics DepartmentofMathematics
&Statistics &Statistics
8888UniversityDrive 8888UniversityDrive
SimonFraserUniversity SimonFraserUniversity
Burnaby,BC,V5A1S6 Burnaby,BC,V5A1S6
CANADA CANADA
[email protected] [email protected]
BrendanRooney AndreaWeirathmueller
9161TenthAvenue 727WindsorStreet
SimonFraserUniversity UniversityofWestern
Burnaby,BC,V3N2T2 Ontario
CANADA Fredericton,ONE2B4G3
[email protected] [email protected]
Editors-in-Chief
Rédacteurs-en-chef
KarlDilcher
K.Taylor
DepartmentofMathematics
andStatistics
DalhousieUniversity
Halifax,NovaScotiaB3H3J5
Canada
[email protected]
CoverFigure:A“randomwalk”ontheLiouvilleFunction.
ISBN:978-0-387-72125-5 e-ISBN:978-0-387-72126-2
LibraryofCongressControlNumber:2007936622
MathematicsSubjectClassification(2000):11-xx01-xx
(cid:2)c2008SpringeScience+BusinessMedia,LLC
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For Pinot
- P. B.
For my parents, my lovely wife, Shirley, my daughter, Priscilla, and son,
Matthew
- S. C.
For my parents Tom and Katrea
- B. R.
For my family
- A. W.
Preface
SinceitsinclusionasaMillenniumProblem,numerousbookshavebeenwrit-
ten to introduce the Riemann hypothesis to the general public. In an average
local bookstore, it is possible to see titles such as John Derbyshire’s Prime
Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Math-
ematics, Dan Rockmore’s Stalking the Riemann Hypothesis: The Quest to
Find the Hidden Law of Prime Numbers, and Karl Sabbagh’s The Riemann
Hypothesis: The Greatest Unsolved Problem in Mathematics.
This book is a more advanced introduction to the theory surrounding
the Riemann hypothesis. It is a source book, primarily composed of relevant
originalpapers,butalsocontainsacollectionofsignificantresults.Thetextis
suitable for a graduate course or seminar, or simply as a reference for anyone
interested in this extraordinary conjecture.
The material in Part I (Chapters 1-10) is mostly organized into indepen-
dent chapters and one can cover the material in many ways. One possibility
is to jump to Part II and start with the four expository papers in Chapter
11. The reader who is unfamiliar with the basic theory and algorithms used
in the study of the Riemann zeta function may wish to begin with Chapters
2 and 3. The remaining chapters stand on their own and can be covered in
anyorderthereaderfancies(obviouslywithourpreferencebeingfirsttolast).
We have tried to link the material to the original papers in order to facilitate
more in-depth study of the topics presented.
We have divided Part II into two chapters. Chapter 11 consists of four
expository papers on the Riemann hypothesis, while Chapter 12 contains the
original papers that developed the theory surrounding the Riemann hypoth-
esis.
Presumably the Riemann hypothesis is very difficult, and perhaps none
of the current approaches will bear fruit. This makes selecting appropriate
papers problematical. There is simply a lack of profound developments and
attacksonthefullproblem.However,thereisanintimateconnectionbetween
the prime number theorem and the Riemann hypothesis. They are connected
VIII Preface
theoreticallyandhistorically,andtheRiemannhypothesismaybethoughtof
asagrandgeneralizationoftheprimenumbertheorem.Thereisalargebody
of theory on the prime number theorem and a progression of solutions. Thus
wehavechosenvariouspapersthatgiveproofsoftheprimenumbertheorem.
WhiletherehavebeennosuccessfulattacksontheRiemannhypothesis,a
significant body of evidence has been generated in its support. This evidence
is often computational; hence we have included several papers that focus on,
or use computation of, the Riemann zeta function. We have also included
Weil’s proof of the Riemann hypothesis for function fields (Section 12.8) and
the deterministic polynomial primality test of Argawal at al. (Section 12.20).
Acknowledgments.Wewouldliketothankthecommunityofauthors,pub-
lishers, and libraries for their kind permission and assistance in republishing
thepapersincludedinPartII.Inparticular,“OnNewman’sQuickWaytothe
PrimeNumberTheorem”and“PairCorrelationofZerosandPrimesinShort
Intervals” are reprinted with kind permission of Springer Science and Busi-
nessMedia,“ThePairCorrelationofZerosoftheZetaFunction”isreprinted
with kind permission of the American Mathematical Society, and “On the
Difference π(x) − Li(x)” is reprinted with kind permission of the London
Mathematical Society.
We gratefully acknowledge Michael Coons, Karl Dilcher, Ron Ferguson
andAlexavanderWaallformanyusefulcommentsandcorrectionsthathelp
make this a better book.
Contents
Part I Introduction to the Riemann Hypothesis
1 Why This Book............................................ 3
1.1 The Holy Grail.......................................... 3
1.2 Riemann’s Zeta and Liouville’s Lambda .................... 5
1.3 The Prime Number Theorem ............................. 7
2 Analytic Preliminaries..................................... 9
2.1 The Riemann Zeta Function .............................. 9
2.2 Zero-free Region......................................... 16
2.3 Counting the Zeros of ζ(s)................................ 18
2.4 Hardy’s Theorem........................................ 24
3 Algorithms for Calculating ζ(s)............................ 29
3.1 Euler–MacLaurin Summation ............................. 29
3.2 Backlund............................................... 30
3.3 Hardy’s Function........................................ 31
3.4 The Riemann–Siegel Formula ............................. 32
3.5 Gram’s Law ............................................ 33
3.6 Turing ................................................. 34
3.7 The Odlyzko–Sch¨onhage Algorithm ........................ 35
3.8 A Simple Algorithm for the Zeta Function .................. 35
3.9 Further Reading......................................... 36
X Contents
4 Empirical Evidence ........................................ 37
4.1 Verification in an Interval ................................ 37
4.2 A Brief History of Computational Evidence................. 39
4.3 The Riemann Hypothesis and Random Matrices............. 40
4.4 The Skewes Number ..................................... 43
5 Equivalent Statements..................................... 45
5.1 Number-Theoretic Equivalences ........................... 45
5.2 Analytic Equivalences.................................... 49
5.3 Other Equivalences ...................................... 52
6 Extensions of the Riemann Hypothesis .................... 55
6.1 The Riemann Hypothesis................................. 55
6.2 The Generalized Riemann Hypothesis...................... 56
6.3 The Extended Riemann Hypothesis........................ 57
6.4 An Equivalent Extended Riemann Hypothesis............... 57
6.5 Another Extended Riemann Hypothesis .................... 58
6.6 The Grand Riemann Hypothesis........................... 58
7 Assuming the Riemann Hypothesis and Its Extensions ... . 61
7.1 Another Proof of The Prime Number Theorem.............. 61
7.2 Goldbach’s Conjecture ................................... 62
7.3 More Goldbach.......................................... 62
7.4 Primes in a Given Interval................................ 63
7.5 The Least Prime in Arithmetic Progressions ................ 63
7.6 Primality Testing........................................ 63
7.7 Artin’s Primitive Root Conjecture ......................... 64
7.8 Bounds on Dirichlet L-Series.............................. 64
7.9 The Lindel¨of Hypothesis.................................. 65
7.10 Titchmarsh’s S(T) Function .............................. 65
7.11 Mean Values of ζ(s) ..................................... 66
8 Failed Attempts at Proof .................................. 69
8.1 Stieltjes and Mertens’ Conjecture.......................... 69
8.2 Hans Rademacher and False Hopes ........................ 70
8.3 Tur´an’s Condition ....................................... 71
8.4 Louis de Branges’s Approach.............................. 71
8.5 No Really Good Idea .................................... 72