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 these formulae, we derive explicit representations for the sum b a n − s for certain ranges of a and b. In addition, we present precise estimates relating this sum with the sum d c n s − 1 for certain ranges of a, b, c, d. We also study a two-parameter 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).
Author(s): Athanassios S. Fokas, Jonatan Lenells
Series: Memoirs of the American Mathematical Society, 1351
Publisher: American Mathematical Society
Year: 2022
Language: English
Pages: 127
City: Providence
Cover
Title page
Part 1. Asymptotics to all Orders of the Riemann Zeta Function
Chapter 1. Introduction
1.1. The large ? asymptotics of ?(?) valid to all orders
1.2. The explicit form of certain sums
1.3. The explicit form of the difference of certain sums
1.4. Asymptotics of a two-parameter generalization of Riemann’s zeta function
1.5. Fourier coefficients of the product of two Hurwitz zeta functions
1.6. Several representations for the basic sum
Chapter 2. An Exact Representation for ?(?)
Chapter 3. The Asymptotics of the Riemann Zeta Function for ?≤?<∞
Chapter 4. The Asymptotics of the Riemann Zeta Function for 0
Chapter 5. Consequences of the Asymptotic Formulae
Part 2. Asymptotics to all Orders of a Two-Parameter Generalization of the Riemann Zeta Function
Chapter 6. An Exact Representation for Φ(?,?,?)
Chapter 7. The Asymptotics of Φ(?,?,?)
Chapter 8. More Explicit Asymptotics of Φ(?,?,?)
Chapter 9. Fourier coefficients of the product of two Hurwitz zeta functions
9.1. Corollaries
9.2. Asymptotics of the zeroth Fourier coefficient
9.3. Proof of Theorem 9.1
9.4. Proof of Theorem 9.2
9.5. Fourier coefficients of ?(?,?) and ?₁(?,?)
Part 3. Representations for the Basic Sum
Chapter 10. Several Representations for the Basic Sum
Appendix A. The Asymptotics of Γ(1-?) and ?(?)
Appendix B. Numerical Verifications
B.1. Verification of Theorem 3.1
B.2. Verification of Theorem 3.2
B.3. Verification of Theorem 4.1
B.4. Verification of Corollary 4.3
B.5. Verification of Theorem 4.4
Bibliography
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