Counting zeros of the Riemann zeta function

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 t...

Full description

Saved in:
Bibliographic Details
Published inarXiv.org
Main Authors Hasanalizade, Elchin, Shen, Quanli, Peng-Jie, Wong
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 14.07.2021
Subjects
Dirichlet problem
Online AccessGet full text

Cover

More Information
Summary: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.
ISSN:2331-8422