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

@article{Jerby2021AnAF,
title={An approximate functional equation for the Riemann zeta function with exponentially decaying error},
author={Yochay Jerby},
journal={J. Approx. Theory},
year={2021},
volume={265},
pages={105551}
}
• Yochay Jerby
• Published 2021
• Computer Science, Mathematics
• J. Approx. Theory
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… Expand

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