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 $γ_1+γ_2$ of two ordinates of nontrivial zeros of the Riemann zeta function. For the associated pair correlation function we prove that $G_2(α,T)=\{\log T/T^{2α}+4α^3/(3T^α)\} \{1+O(1/\log\log T)\}$ uniformly for $0\leα\le 2/3-2\log\log T/\log T$. 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.