Counting zeros of the Riemann zeta function

TL;DR

This paper provides an improved explicit bound on the number of non-trivial zeros of the Riemann zeta function up to height T, utilizing advanced bounds and techniques from related L-function studies.

Contribution

It introduces a tighter explicit bound on zero counting of the Riemann zeta function, improving upon previous results for large T.

Findings

New explicit bound with smaller constants

Utilizes subconvexity bounds and techniques from Dirichlet L-functions

Enhances previous zero counting estimates

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.