Improved estimates for the argument and zero-counting function of the Riemann zeta-function

TL;DR

This paper refines the estimates for the zero-counting function of the Riemann zeta-function by applying new subconvexity bounds, leading to tighter bounds on the number of zeros up to height T.

Contribution

It introduces improved bounds for the zero-counting function of the Riemann zeta-function using novel subconvexity estimates within the critical strip.

Findings

Established a new explicit bound for N(T) with smaller constants.

Implemented subconvexity bounds to enhance zero-counting estimates.

Achieved tighter error terms in the zero-counting function.

Abstract

In this article, we improve the recent work of Hasanalizade, Shen, and Wong by establishing \[ \left| N (T) - \frac{T}{ 2 \pi} \log \left( \frac{T}{2\pi e}\right) \right|\le 0.10076\log T+0.24460\log\log T+8.08344, \] for every $T\ge e$, where $N(T)$ is the number of non-trivial zeros $\rho=\beta+i\gamma$, with $0<\gamma \le T$, of the Riemann zeta-function $\zeta(s)$. The main source of improvement comes from implementing new subconvexity bounds for $\zeta(\sigma+it)$ on some $\sigma_k$-lines inside the critical strip.