Pair correlation for sums of two ordinates of zeros of the Riemann zeta function

Abstract

Assuming the Riemann Hypothesis, we extend Montgomery's pair correlation method to study the distribution of differences between sums ${\gamma }_1+{\gamma }_2$ of two ordinates of nontrivial zeros of the Riemann zeta function. For the associated pair correlation function we prove that $G_2(\alpha ,T)=\{\log T/T^{2\alpha }+4{\alpha }^3/(3T^{\alpha })\}\{1+O(1/\log \log T)\}$ uniformly for $0\le \alpha \le 2/3-2\log \log T/\log T$. In contrast with the conjectured GUE statistics of the ordinates themselves, this result points to an absence of level repulsion among sums of two ordinates, the two-point correlation function of the sums being identically one, as for a Poisson process.

16 pages. v2 is a major revision with a new title (v1: "Pair correlation: Variations on a theme"). This version treats only pair correlation for sums of two ordinates, proving a stronger and more uniform asymptotic than the corresponding case mu=2 of v1. The proofs of Theorems 1.2 and 1.3 of v1 for general mu contain gaps, and those statements are withdrawn. Comments welcome