The third moment of the logarithm of zeta and a twisted pair correlation conjecture

TL;DR

This paper provides refined conditional estimates for the third moment of the logarithm of the Riemann zeta function, aligning with predictions and relying on several unproven conjectures including the Riemann Hypothesis.

Contribution

It introduces new conditional estimates for the third moment of the zeta function's logarithm, assuming multiple conjectures, and justifies a twisted pair correlation conjecture.

Findings

Conditional estimates match Keating and Snaith's predictions

Proves the twisted pair correlation conjecture unconditionally in a certain range

Extends results under a Hardy-Littlewood conjecture variant

Abstract

We prove precise conditional estimates for the third moment of the logarithm of the Riemann zeta function, refining what is implied by the Selberg central limit theorem, both for the real and imaginary parts. These estimates match predictions made in work of Keating and Snaith. We require the Riemann Hypothesis, a conjecture for the triple correlation of Riemann zeros and another ``twisted'' pair correlation conjecture which explains the interaction of a prime power with Montgomery's pair correlation function. We believe this to be of independent interest, and devote substantial effort to its justification. Namely, we prove this conjecture on a certain range unconditionally, and on a larger range under the assumption of a variant of the Hardy-Littlewood conjecture with good uniformity.