Zero-free half-planes of the \zeta -function via spaces of analytic functions

TL;DR

This paper develops a new method using spaces of analytic functions to identify zero-free regions of the Riemann zeta function, providing precise conditions for such half-planes.

Contribution

It introduces a general approach linking topological vector spaces of analytic functions to zero-free regions of , applied to weighted  and Hardy spaces.

Findings

Derived conditions for zero-free half-planes of 

Applied method to weighted  and Hardy spaces

Established a framework for analyzing  zeros

Abstract

In this article, we introduce a general approach for deriving zero-free half-planes for the Riemann zeta function $\zeta$ by identifying topological vector spaces of analytic functions with specific properties. This approach is applied to weighted $\ell^2$ spaces and classical Hardy spaces $ H^p $ ($ 0<p\leq2 $). As a consequence precise conditions are obtained for the existence of zero-free half planes for the $\zeta$-function.