Pair correlation: Variations on a theme

TL;DR

This work extends Montgomery's pair correlation framework in two directions: (i) a two-parameter family of real weights $w_{\nu,k}(u)$ to probe the Fourier-analytic structure of zero differences and (ii) a generalized pair correlation for sums of ordinates $\Sigma\gamma$ of zeros. The authors establish explicit asymptotics for the weighted correlation $F_{\nu,k}(\alpha)$ under RH, revealing a delta-like term and a secondary $\alpha$-dependent correction that is universal across the family, and they propose a conjecture $F_{\nu,k}(\alpha)=1+o(1)$ on bounded $\alpha$-ranges, which would imply PCC for all choices of $\nu,k$. They also develop a parallel theory for ordinate sums $G_\mu(\alpha)$, proving sharp asymptotics for $\mu=2,3$ and uniform bounds for $\mu\ge 4$, while formulating a broad SPC framework ${\tt SPC}[\cH_\mu]$ and conjectures ${\tt PCC}[\mu]$ that extend PCC to higher-order sum structures. Assuming LIC, they show how these SPC conjectures imply precise counting results for near-diagonal ordinate sums and yield PCC-type laws for the sums of $\mu$ ordinates, connecting zero multiplicities, linear independence, and the fine-scale statistics of zeta zeros. Overall, the paper highlights that while different weights yield distinct asymptotics, they do not uncover new information about simple zeros, whereas the higher-order ordinate-sum correlations exhibit strong, quantifiable order under plausible conjectures, with potential implications for the deep structure of zeta zeros.

Abstract

In his groundbreaking work on pair correlation, Montgomery analyzed the distribution of the differences $γ'-γ$ between ordinates $γ$ 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 $Σγ=γ_1+\cdots+γ_μ$. Our results suggest a natural generalization of the pair correlation conjecture for any integer $μ\ge 2$.

Figures (1)

• Figure 1: Weighting functions $w_{1,j}$ with $j\in\{0,1,2,3\}$.

Theorems & Definitions (29)

• theorem 1
• theorem 2: Asymptotic estimates
• theorem 3: Upper bounds
• lemma 1
• lemma 2
• proof : Proof sketch
• lemma 3
• proof : Proof sketch
• lemma 4
• lemma 5