On the Asymptotics to all Orders of the Riemann Zeta Function and of a Two-Parameter Generalization of the Riemann Zeta Function

128 Pages·2022·1.06 MB·English

by  Athanassios S. Fokas, Jonatan Lenells

Number 1351
On the Asymptotics to all Orders of
the Riemann Zeta Function and of a
Two-Parameter Generalization of the
Riemann Zeta Function
Athanassios S. Fokas
Jonatan Lenells
January 2022 • Volume 275 • Number 1351 (fifth of 6 numbers)
Number 1351
On the Asymptotics to all Orders of
the Riemann Zeta Function and of a
Two-Parameter Generalization of the
Riemann Zeta Function
Athanassios S. Fokas
Jonatan Lenells
January 2022 • Volume 275 • Number 1351 (fifth of 6 numbers)
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DOI:https://doi.org/10.1090/memo/1351
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Contents
Part 1. Asymptotics to all Orders of the Riemann Zeta Function 1
Chapter 1. Introduction 3
1.1. The large t asymptotics of ζ(s) valid to all orders 3
1.2. The explicit form of certain sums 5
1.3. The explicit form of the difference of certain sums 6
1.4. Asymptotics of a two-parameter generalization of Riemann’s zeta
function 7
1.5. Fourier coefficients of the product of two Hurwitz zeta functions 7
1.6. Several representations for the basic sum 9
Chapter 2. An Exact Representation for ζ(s) 11
Chapter 3. The Asymptotics of the Riemann Zeta Function for t≤η <∞ 15
Chapter 4. The Asymptotics of the Riemann Zeta Function for 0<η <t 31
Chapter 5. Consequences of the Asymptotic Formulae 49
Part 2. Asymptotics to all Orders of a Two-Parameter
Generalization of the Riemann Zeta Function 53
Chapter 6. An Exact Representation for Φ(u,v,β) 55
Chapter 7. The Asymptotics of Φ(u,v,β) 59
Chapter 8. More Explicit Asymptotics of Φ(u,v,β) 73
Chapter 9. Fourier coefficients of the product of two Hurwitz zeta functions 81
9.1. Corollaries 83
9.2. Asymptotics of the zeroth Fourier coefficient 86
9.3. Proof of Theorem 9.1 89
9.4. Proof of Theorem 9.2 91
9.5. Fourier coefficients of ζ(s,α) and ζ (s,α) 93
1
Part 3. Representations for the Basic Sum 95
Chapter 10. Several Representations for the Basic Sum 97
Appendix A. The Asymptotics of Γ(1−s) and χ(s) 107
Appendix B. Numerical Verifications 109
B.1. Verification of Theorem 3.1 109
v
vi CONTENTS
B.2. Verification of Theorem 3.2 109
B.3. Verification of Theorem 4.1 110
B.4. Verification of Corollary 4.3 110
B.5. Verification of Theorem 4.4 110
Bibliography 113
Abstract
We present several formulae for the large t asymptotics of the Riemann zeta
function ζ(s), s = σ +it, 0 ≤ σ ≤ 1, t > 0, which are valid to all orders. A
particular case of these results coincides with the classical results of Siegel. Using
(cid:2)
these formulae, we derive explicit representations for the sum bn−s for certain
a
ranges of a and b. In addition, we present precise estimates relating this sum with
(cid:2)
the sum dns−1 for certain ranges of a,b,c,d. We also study a two-parameter
c
generalization of the Riemann zeta function which we denote by Φ(u,v,β), u∈C,
v ∈ C, β ∈ R. Generalizing the methodology used in the study of ζ(s), we derive
asymptotic formulae for Φ(u,v,β).
Received by the editor December 1, 2015, and, in revised form, July 6, 2018, October 26,
2018,andNovember5,2018.
ArticleelectronicallypublishedonDecember21,2021.
DOI:https://doi.org/10.1090/memo/1351
2020 MathematicsSubjectClassification. Primary11M06,30E15;Secondary33E20.
Key wordsand phrases. Riemannzetafunction,asymptotics.
The authors are grateful to Professor D. Bump for several useful suggestions that led to a
significant improvement of the manuscript. They are also grateful to Kostis Kalimeris for many
helpfulremarks.
The first author is affiliated with the Department of Applied Mathematics and Theoretical
Physics,UniversityofCambridge,Cambridge,CB30WA,UnitedKingdomandwassupportedby
the EPSRC, UK via a senior fellowship and by the Guggenheim Foundation, USA. He is deeply
grateful to his students Michail Dimakos (partially supported by the Onassis Foundation) and
Dionyssis Mantzavinos (partially supported by the EPSRC, UK) for performing a large number
of numerical experiments during the early stages of this project. He also expresses his sincere
gratitudetoAnthonyAshton,BryceMcLeod,JoeKeller,DavidStuart,andEugeneShargorodsky,
forextensivediscussionsandimportantsuggestions.
The second author is affiliated with the Department of Mathematics, KTH Royal Institute
of Technology, 100 44 Stockholm, Sweden and acknowledges support from the EPSRC, UK, the
G¨oran Gustafsson Foundation, Sweden, the Ruth and Nils-Erik Stenb¨ack Foundation, Finland,
theEuropeanResearchCouncil,GrantAgreementNo. 682537,andtheSwedishResearchCouncil,
GrantNo. 2015-05430.
(cid:2)c2022 American Mathematical Society
vii

Part 1
Asymptotics to all Orders of the
Riemann Zeta Function