Numerical Computations Concerning the GRH

TL;DR

This paper introduces two new algorithms for computing Dirichlet L-functions efficiently and rigorously, enabling verification of the Generalised Riemann Hypothesis for all primitive characters with modulus up to 400,000.

Contribution

The paper presents novel algorithms that significantly improve the computational verification of the GRH for a large class of L-functions.

Findings

Verified GRH for all primitive characters with q<=400,000 up to specified heights.

Achieved efficient and rigorous computation of Dirichlet L-functions.

Extended the computational verification to a large range of moduli.

Abstract

We describe two new algorithms for the efficient and rigorous computation of Dirichlet L-functions and their use to verify the Generalised Riemann Hypothesis for all such L-functions associated with primitive characters of modulus q<=400,000. For even q, we check to height t_0=max(1e8/q,7.5e7/q+200) and for odd q to height t_0=max(1e8/q,3.75e7/q+200).