A remark on density theorems for Riemann's zeta-function

TL;DR

This paper provides a straightforward proof of existing zero density estimates for the Riemann zeta function, enabling progress beyond the density hypothesis in parts of the critical strip using classical methods.

Contribution

It offers a simplified proof of known zero density estimates that surpass the density hypothesis in certain regions, relying solely on classical results.

Findings

Zero density estimates are strengthened in parts of the critical strip.

The proof simplifies previous approaches by using classical knowledge.

It breaks the density hypothesis in a nontrivial region.

Abstract

The goal of this paper is to give a relatively simple proof of some known zero density estimates for Riemann zeta function which are sufficiently strong to break the density hypothesis in a nontrivial part of the critical strip. Apart from a simple but ingenious idea of Halasz the proof uses only classical knowledge about the zeta function, results known since at least hundred years.