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arXiv · 2502.20569

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

William D. Banks
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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.

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William D. Banks. 2025-02-27. Pair correlation for sums of two ordinates of zeros of the Riemann zeta function. https://arxiv.org/abs/2502.20569

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