Sharp conditional moment bounds for products of $L$-functions

TL;DR

This paper establishes sharp bounds for moments of products of $L$-functions on the critical line under GRH and GRC, providing insights into their independence and extending to Hurwitz zeta and Dedekind zeta functions.

Contribution

It introduces conditional sharp moment bounds for products of $L$-functions, advancing understanding of their behavior and independence under key hypotheses.

Findings

Conditional order of moments for products of $L$-functions determined

Sharp bounds for Hurwitz zeta functions with rational parameters obtained

Results imply conditional independence of $L$-functions in large deviations regime

Abstract

Assuming the Generalized Riemann Hypothesis and the Generalized Ramanujan Conjecture, we determine the order of the $2(k_1,\dots,k_r)$th moment of a product of distinct irreducible $L$-functions on the critical line. As a consequence, we obtain conditional information about the independence of these $L$-functions in the large deviations regime. We also obtain sharp moment bounds for Hurwitz zeta functions with rational parameter, and a certain family of Dedekind zeta functions.