Macroscopic pair correlation of the Riemann zeroes for smooth test functions

TL;DR

Under the Riemann hypothesis, this paper proves that the pair correlation measure of Riemann zeta zeroes closely matches a conjectured measure for smooth test functions, extending Montgomery's GUE result.

Contribution

It extends Montgomery's pair correlation result to a broader class of smooth test functions, aligning the measure with Bogomolny and Keating's conjecture.

Findings

Pair correlation measure matches the conjectured measure with small error.

Detection of macroscopic troughs in the pair correlation measure.

Results hold under the Riemann hypothesis for smooth test functions.

Abstract

On the assumption of the Riemann hypothesis, we show that over a class of sufficiently smooth test functions, a measure conjectured by Bogomolny and Keating coincides to a very small error with the actual pair correlation measure for zeroes of the Riemann zeta function. Our result extends the well known result of Montgomery that over the same class of test functions the pair correlation measure coincides (to a larger error term) with that of the Gaussian Unitary Ensemble (GUE). The restriction of test functions remains stringent, but we are nonetheless able to detect, at a microscopically blurred resolution, macroscopic troughs in the pair correlation measure.