The Zeros of Zeta Function Revisited

Zhengjun Cao; Lihua Liu

Paper 2024/1412

The Zeros of Zeta Function Revisited

Zhengjun Cao

Lihua Liu

Abstract

Let $\zeta(z)=\sum_{n=1}^{\infty} \frac{1}{n^z}$, $\psi(z)=\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^z}, z\in \mathbb{C}$. We show that $\psi(z)\not=(1-2^{1-z})\zeta(z)$, if $0<z<1$. Besides, we clarify that the known zeros are not for the original series, but very probably for the alternating series.

Metadata

Available format(s): PDF
Category: Foundations
Publication info: Preprint.
Keywords: Zeta function alternating series partial sum absolute convergence
Contact author(s): liulh @ shmtu edu cn
History: 2024-09-11: approved; 2024-09-10: received; See all versions
Short URL: https://ia.cr/2024/1412
License: CC0

BibTeX

@misc{cryptoeprint:2024/1412,
      author = {Zhengjun Cao and Lihua Liu},
      title = {The Zeros of Zeta Function Revisited},
      howpublished = {Cryptology {ePrint} Archive, Paper 2024/1412},
      year = {2024},
      url = {https://eprint.iacr.org/2024/1412}
}