The Riemann Zeta-Function and the Sine Kernel

TL;DR

This paper explores the connection between the sine kernel and shifted moments of the Riemann zeta function on the critical line, providing rigorous results for second and fourth moments and conjecturing a broader link for higher moments.

Contribution

It introduces a novel connection between the sine kernel and the moments of the Riemann zeta function, extending known results and proposing a new conjecture relating to higher moments.

Findings

Rigorous results for the second and fourth shifted moments.

Conjecture linking the sine kernel to higher even moments.

Connection to a recent conjecture by Conrey et al.

Abstract

We point out an interesting occurrence of the sine kernel in connection with the shifted moments of the Riemann zeta function along the critical line. We discuss rigorous results in this direction for the shifted second moment and for the shifted fourth moment. Furthermore, we conjecture that the sine kernel also occurs in connection with the higher (even) shifted moments and show that this conjecture is closely related to a recent conjecture by Conrey, Farmer, Keating, Rubinstein, and Snaith.