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On the Asymptotics to all Orders of the Riemann Zeta Function and of a Two-Parameter Generalization of the Riemann Zeta Function
About this Title
Athanassios S. Fokas and Jonatan Lenells
Publication: Memoirs of the American Mathematical Society
Publication Year:
2022; Volume 275, Number 1351
ISBNs: 978-1-4704-5098-4 (print); 978-1-4704-7017-3 (online)
DOI: https://doi.org/10.1090/memo/1351
Published electronically: December 21, 2021
Keywords: Riemann zeta function,
asymptotics
Table of Contents
1. Asymptotics to all Orders of the Riemann Zeta Function
- 1. Introduction
- 2. An Exact Representation for $\zeta (s)$
- 3. The Asymptotics of the Riemann Zeta Function for $t \leq \eta < \infty$
- 4. The Asymptotics of the Riemann Zeta Function for $0 < \eta < t$
- 5. Consequences of the Asymptotic Formulae
2. Asymptotics to all Orders of a Two-Parameter Generalization of the Riemann Zeta Function
- 6. An Exact Representation for $\Phi (u,v,\beta )$
- 7. The Asymptotics of $\Phi (u,v,\beta )$
- 8. More Explicit Asymptotics of $\Phi (u,v, \beta )$
- 9. Fourier coefficients of the product of two Hurwitz zeta functions
3. Representations for the Basic Sum
- 10. Several Representations for the Basic Sum
- A. The Asymptotics of $\Gamma (1-s)$ and $\chi (s)$
- B. Numerical Verifications
Abstract
We present several formulae for the large $t$ asymptotics of the Riemann zeta function $\zeta (s)$, $s=\sigma +i t$, $0\leq \sigma \leq 1$, $t>0$, 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 $\sum _a^b n^{-s}$ for certain ranges of $a$ and $b$. In addition, we present precise estimates relating this sum with the sum $\sum _c^d n^{s-1}$ for certain ranges of $a, b, c, d$. We also study a two-parameter generalization of the Riemann zeta function which we denote by $\Phi (u,v,\beta )$, $u\in \mathbb {C}$, $v\in \mathbb {C}$, $\beta \in \mathbb {R}$. Generalizing the methodology used in the study of $\zeta (s)$, we derive asymptotic formulae for $\Phi (u,v, \beta )$.- Johan Andersson, Mean value properties of the Hurwitz zeta-function, Math. Scand. 71 (1992), no. 2, 295–300. MR 1212712, DOI 10.7146/math.scand.a-12430
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