Moments of the Dedekind zeta function and other non-primitive L-functions

TL;DR

This paper proposes a conjecture for the moments of Dedekind zeta functions and other non-primitive L-functions, combining heuristic and proven results, with applications to quadratic extensions and generalizations.

Contribution

It introduces a new conjecture for moments of Dedekind zeta functions using hybrid methods and extends these ideas to non-primitive L-functions.

Findings

Conjectured moments for Dedekind zeta functions of Galois extensions.

Proved asymptotic for the first moment in quadratic extensions.

Generalized methods to non-primitive L-functions.

Abstract

We give a conjecture for the moments of the Dedekind zeta function of a Galois extension via the hybrid product method. The moments of the product of primes are evaluated using the Montgomery-Vaughan mean value theorem whilst for the moments of the product over zeros we give a heuristic argument involving random matrix theory. The asymptotic for the first moment of the product over zeros is then proved for quadratic extensions. We are also able to reproduce our main conjecture in the quadratic case by using a modified version of the moments recipe. Finally, we generalise our methods to give a conjecture for moments of non-primitive L-functions.