A note on Hardy's theorem

TL;DR

This paper presents a new, simplified proof of Hardy's theorem, which states that the Riemann zeta-function has infinitely many zeros on the critical line, contributing to the understanding of the distribution of these zeros.

Contribution

It provides a novel, straightforward proof of Hardy's theorem, enhancing the mathematical toolkit for studying zeros of the Riemann zeta-function.

Findings

Confirmed infinitely many zeros on the critical line

Introduced a simpler proof technique

Potentially facilitates further research on zeta zeros

Abstract

Hardy's theorem for the Riemann zeta-function $\zeta(s)$ says that it admits infinitely many complex zeros on the line $\Re({s}) = \frac{1}{2}$. In this note, we give a simple proof of this statement which, to the best of our knowledge, is new.