Pair Correlation of Zeros of the Riemann Zeta Function I: Proportions of Simple Zeros and Critical Zeros

TL;DR

This paper extends Montgomery's pair correlation method to analyze the distribution of zeros of the Riemann zeta-function, showing that at least two-thirds are simple and on the critical line under certain conditions.

Contribution

It demonstrates that pair correlation techniques can be applied to the horizontal distribution of zeros, not just their pairwise spacing, under weaker assumptions than RH.

Findings

At least 2/3 of zeros are simple.

At least 2/3 of zeros lie on the critical line.

At least 1/3 of zeros are simple and on the critical line.

Abstract

Assuming the Riemann Hypothesis (RH), Montgomery proved a theorem in 1973 concerning the pair correlation of zeros of the Riemann zeta-function and applied this to prove that at least $2/3$ of the zeros are simple. In this paper, we investigate the versatility of the pair correlation method and show, for the first time, that it can be used to prove results on the \emph{horizontal distribution} of zeros of the Riemann zeta-function. In earlier work we showed how to remove RH from Montgomery's theorem and, in turn, obtain results on simple zeros assuming conditions on the zeros that are weaker than RH. Here we assume a more general condition, namely that all the zeros $\rho = \beta +i\gamma$ with $T<\gamma\le 2T$ are in a narrow vertical box centered on the critical line with width $b/\log T$, where $b\to 0$ as $T\to \infty$. We first prove the generalization of Montgomery's result that at least $2/3$ of zeros are simple, and we then prove the new result that the pair correlation method yields at least $2/3$ of the zeros on the critical line. We also use the pair correlation method to prove that at least $1/3$ of the zeros are both simple and on the critical line, a result already known unconditionally using different methods.