An approximate functional equation for the Riemann zeta function with exponentially decaying error

TL;DR

This paper derives a new approximate functional equation for the Riemann zeta function using the Hasse-Sondow series, featuring an exponentially decaying error term, which improves the understanding of its asymptotic properties.

Contribution

It introduces an approximate functional equation for the Riemann zeta function with an exponentially decaying error term, based on saddle point analysis of associated integral representations.

Findings

Error term decays exponentially with respect to parameters x and y

Provides asymptotic analysis of the function (u,s) using saddle point techniques

Enhances the precision of the Riemann zeta function approximation

Abstract

It is known by a formula of Hasse-Sondow that the Riemann zeta function is given, for any $ s=\sigma+it \in \mathbb{C}$, by $ \sum_{n=0}^{\infty} \widetilde{A}(n,s)$ where $$ \widetilde{A}(n,s):=\frac{1}{2^{n+1}(1-2^{1-s})} \sum_{k=0}^n \binom{n}{k}\frac{(-1)^k}{(k+1)^s} .$$ We prove the following approximate functional equation for the Hasse-Sondow presentation: For $ \vert t \vert = \pi xy $ and $ 2y \neq (2N-1)\pi $ then $$ \zeta(s)= \sum_{n \leq x } \widetilde{A}(n,s)+\frac{\chi(s)}{1-2^{s-1}} \left (\sum_{k \leq y} (2k-1)^{s-1} \right ) +O \left (e^{-\omega(x,y) t} \right ), $$ where $ 0 <\omega(x,y)$ is a certain transcendental number determined by $ x$ and $ y$. A central feature of our new approximate functional equation is that its error term is of exponential rate of decay. The proof is based on a study, via saddle point techniques, of the asymptotic properties of the function $$ \widetilde{A}(u,s):= \frac{1}{2^{u+1} (1-2^{1-s}) \Gamma(s)} \int_{0}^{\infty} \left ( e^{-w} \left ( 1- e^{-w} \right)^u \right ) w^{s-1} dw,$$ and integrals related to it.