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

TL;DR

This paper develops all-order asymptotic formulas for the Riemann zeta function and a two-parameter generalization, providing explicit representations and estimates, extending classical results and introducing new analytical tools.

Contribution

It introduces comprehensive all-order asymptotic formulas for ta and a novel two-parameter generalization, ta, with explicit representations and estimates.

Findings

Derived formulas valid to all orders for ta asymptotics.

Established explicit sum representations and estimates for ta.

Extended classical asymptotic results to a new two-parameter function.

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 $\Phi(u,v,\beta)$.