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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