A method for proving the completeness of a list of zeros of certain L-functions

TL;DR

This paper introduces a generalized explicit formula-based method for verifying the completeness of zeros of various L-functions, improving efficiency over previous approaches like Turing's method.

Contribution

It extends the Weil-Barner explicit formula approach to a broad class of L-functions, including Hecke, Artin, and automorphic L-functions, with specific application to elliptic curves.

Findings

The method asymptotically sacrifices fewer zeros than Turing's method.

It applies to a wide range of L-functions beyond the Riemann zeta function.

Demonstrated effectiveness on Hecke L-series and elliptic curve L-functions.

Abstract

When it comes to partial numerical verification of the Riemann Hypothesis, one crucial part is to verify the completeness of a list of pre-computed zeros. Turing developed such a method, based on an explicit version of a theorem of Littlewood on the average of the argument of the Riemann zeta function. In a previous paper we suggested an alternative method based on the Weil-Barner explicit formula. This method asymptotically sacrifices fewer zeros in order to prove the completeness of a list of zeros with imaginary part in a given interval. In this paper, we prove a general version of this method for an extension of the Selberg class including Hecke and Artin L-series, L-functions of modular forms, and, at least in the unramified case, automorphic L-functions. As an example, we further specify this method for Hecke L-series and L-functions of elliptic curves over the rational numbers.