Pair correlation: Variations on a theme

TL;DR

This paper extends Montgomery's pair correlation analysis of zeta zeros by introducing new weighting functions and exploring the distribution of sums of zeros, proposing a broader conjecture for multiple zeros.

Contribution

It introduces a two-parameter family of weights and generalizes the pair correlation conjecture to sums of multiple zeros.

Findings

New weighting functions alter asymptotic behavior.

No new information about zero simplicity from weights.

Proposes a generalized conjecture for sums of zeros.

Abstract

In his groundbreaking work on pair correlation, Montgomery analyzed the distribution of the differences $\gamma'-\gamma$ between ordinates $\gamma$ of the nontrivial zeros of the Riemann zeta function, assuming the Riemann Hypothesis. In this paper, we extend his ideas along two distinct directions. First, we introduce an infinite two-parameter family of real weighting functions that generalize Montgomery's original weight $w$. Although these new weights give rise to pair correlation functions with different asymptotic behavior, they do not yield any new information about the simplicity of the zeros of the zeta function. Second, we extend Montgomery's approach to study the distribution of ordinate sums of the form $\Sigma\gamma=\gamma_1+\cdots+\gamma_\mu$. Our results suggest a natural generalization of the pair correlation conjecture for any integer $\mu\ge 2$.