Correlations of the squares of the Riemann zeta on the critical line

TL;DR

This paper computes the average of products of squared Riemann zeta values on the critical line with large shifts, providing explicit formulas and spectral expansions, and partially verifies a conjecture related to zeta moments.

Contribution

It introduces explicit formulas and spectral expansions for averages of shifted squares of the Riemann zeta, advancing understanding of its moments and correlations.

Findings

Derived an explicit expression for the average of shifted squares of zeta

Provided an approximate spectral expansion for the error term

Partially verified and refuted aspects of Bailey and Keating's conjecture

Abstract

We compute the average of a product of two shifted squares of the Riemann zeta on the critical line with shifts up to size $T^{3/2-\varepsilon}$. We give an explicit expression for such an average and derive an approximate spectral expansion for the error term similar to Motohashi's. As a consequence, we also compute the $(2,2)$-moment of moment of the Riemann zeta, for which we partially verify (and partially refute) a conjecture of Bailey and Keating.