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

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 ${\sum }_{n=0}^{\infty }\tilde{A}(n,s)$ where $$\tilde{A}(n,s):=\frac{1}{2^{n+1}(1-2^{1-s})}\sum _{k=0}^n\left(\genfrac{}{}{0ex}{}{n}{k}\right)\frac{(-1{)}^k}{(k+1{)}^s}.$$ We prove the following approximate functional equation for the Hasse-Sondow presentation: For $|t|=\pi xy$ and $2y\ne (2N-1)\pi$ then $$\zeta (s)=\sum _{n\le x}\tilde{A}(n,s)+\frac{\chi (s)}{1-2^{s-1}}\left(\sum _{k\le y}(2k-1{)}^{s-1}\right)+O\left(e^{-\omega (x,y)t}\right),$$ 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 $$\tilde{A}(u,s):=\frac{1}{2^{u+1}(1-2^{1-s})\Gamma (s)}{\int }_0^{\infty }\left(e^{-w}{\left(1-e^{-w}\right)}^u\right)w^{s-1}dw,$$ and integrals related to it.