Counting zeros of the Riemann zeta function

Elchin Hasanalizade; Quanli Shen; Peng-Jie Wong

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arXiv · 2107.06506

Counting zeros of the Riemann zeta function

Elchin Hasanalizade·Quanli Shen·Peng-Jie Wong

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Abstract

In this article, we show that $$ \left| N (T) - \frac{T}{ 2 \pi} \log \left( \frac{T}{2\pi e}\right) \right| \le 0.1038 \log T + 0.2573 \log\log T + 9.3675 $$ where $N(T)$ denotes the number of non-trivial zeros $\rho$, with $0<\Im(\rho) \le T$, of the Riemann zeta function. This improves the previous result of Trudgian for sufficiently large $T$. The improvement comes from the use of various subconvexity bounds and ideas from the work of Bennett $et$ $al.$ on counting zeros of Dirichlet $L$-functions.

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Elchin Hasanalizade, Quanli Shen, Peng-Jie Wong. 2021-07-14. Counting zeros of the Riemann zeta function. https://arxiv.org/abs/2107.06506

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