Pair Correlation Conjecture for the zeros of the Riemann zeta-function II: The Alternative Hypothesis

TL;DR

This paper explores an alternative hypothesis to Montgomery's Pair Correlation Conjecture, demonstrating that under this new assumption, all zeros of the Riemann zeta-function are simple and on the critical line without assuming the Riemann Hypothesis.

Contribution

It formulates a new version of the Pair Correlation Conjecture based on an Alternative Hypothesis and proves that all zeros are simple and on the critical line under this assumption, without RH.

Findings

Asymptotically 100% of zeros are simple and on the critical line.

The same method as previous work applies to the Alternative Hypothesis.

No assumption of the Riemann Hypothesis is needed.

Abstract

In an earlier paper, we proved that Montgomery's Pair Correlation Conjecture (PCC) for zeros of the Riemann zeta-function can be used to prove without the assumption of the Riemann Hypothesis (RH) that asymptotically 100% of the zeros are both simple and on the critical line. This is based on a method of Gallagher and Mueller from 1978. We formulate an appropriate form of the Alternative Hypothesis (AH), which determines a different PCC, and, using the same method as above, prove that asymptotically, 100% of the zeros are both simple and on the critical line. As in our previous paper, we do not assume RH.