Improvement of the theorem of Hardy-Littlewood on density of zeros of the function $\zeta(1/2+it)$

TL;DR

This paper enhances the Hardy-Littlewood theorem on the density of zeros of the Riemann zeta function, reducing the exponent by approximately 16.6%, which is a significant step toward proving Selberg's hypothesis.

Contribution

It introduces a discrete method to improve the classical Hardy-Littlewood exponent, advancing the understanding of zeros of the zeta function.

Findings

Exponent improved by 16.6%

First step toward proving Selberg's hypothesis

Uses novel discrete method

Abstract

In this paper we improve classical Hardy-Littlewood exponent $1/2$ by about $16.6\%$ 62 years after the original result. This result is the first step to prove the Selberg's hypothesis (1942). In order to reach our purpose we use discrete method. This paper is the English version of our paper \cite{4}.