Zeta Zeros on the Critical Line

TL;DR

This paper explores the relationship between the Riemann Hypothesis and the simplicity of zeta zeros, proposing that removing RH assumptions could still prove that most zeros are simple and on the critical line.

Contribution

It demonstrates that eliminating the Riemann Hypothesis assumption from Montgomery's zero simplicity proof still implies most zeros are simple and on the critical line.

Findings

Removing RH from the proof still implies 2/3 of zeros are simple

The approach connects zero simplicity to the critical line distribution

Recent works have applied this idea to other zero-related results

Abstract

Montgomery in 1973 introduced the pair correlation method to study the vertical distribution of Riemann zeta-function zeros. This work assumed the Riemann Hypothesis (RH). One striking application was a short proof that at least 2/3 of zeta-zeros are simple zeros, the first result of its type. Over the last 50 years, most work on pair correlation of zeta-zeros has continued to assume RH. Here we show that if RH could be removed from Montgomery's simple zero proof, then this would also give a proof that 2/3 of the zeros are simple and on the critical line. This idea has been applied in several recent papers to obtain other results on the zeros.