The Weak Gram Law for Hecke $L$-functions

TL;DR

This paper extends Titchmarsh's theorem on Hardy's Z-function to Hecke L-functions, establishing a weak Gram law through contour integration rather than traditional methods, advancing understanding of L-function value distributions.

Contribution

It introduces a novel approach using contour integration to prove the weak Gram law for Hecke L-functions, generalizing previous results for the Riemann zeta function.

Findings

Proves the weak Gram law for Hecke L-functions.

Employs contour integration as a new method.

Generalizes Titchmarsh's theorem to a broader class of L-functions.

Abstract

We generalize a theorem by Titchmarsh about the mean value of Hardy's $Z$-function at the Gram points to the Hecke $L$-functions, which in turn implies the weak Gram law for them. Instead of proceeding analogously to Titchmarsh with an approximate functional equation we employ a different method using contour integration.