Continuous lower bounds for moments of zeta and L-functions

TL;DR

This paper establishes continuous lower bounds for the moments of the Riemann zeta function for all real k > 1, extending previous results limited to rational k and applying to L-functions as well.

Contribution

It introduces a new method that provides continuous lower bounds for moments of zeta and L-functions, covering irrational k values.

Findings

Lower bounds are valid for all real k > 1.

Bounds are of the correct order of magnitude.

Method applies to moments of L-functions in families.

Abstract

We obtain lower bounds of the correct order of magnitude for the 2k-th moment of the Riemann zeta function for all k > 1. Previously such lower bounds were known only for rational values of k, with the bounds depending on the height of the rational number k. Our new bounds are continuous in k, and thus extend also to the case when k is irrational. The method is a refinement of an approach of Rudnick and Soundararajan, and applies also to moments of L-functions in families.