On the zeros of Dirichlet $L$-functions

TL;DR

This paper investigates the zeros of Dirichlet L-functions by computing Li coefficients, verifying their positivity, and proposing a criterion for the partial Riemann hypothesis supported by numerical evidence.

Contribution

It introduces a new numerical approach to verify Li coefficients for Dirichlet L-functions and formulates a criterion related to the partial Riemann hypothesis.

Findings

Positivity of Li coefficients verified for Dirichlet L-functions.

A new criterion for the partial Riemann hypothesis is proposed.

Numerical evidence supports the formulated criterion.

Abstract

In this paper, we compute and verify the positivity of the Li coefficients for the Dirichlet $L$-functions using an arithmetic formula established in Omar and Mazhouda, J. Number Theory 125 (2007) no.1, 50-58; J. Number Theory 130 (2010) no.4, 1109-1114. Furthermore, we formulate a criterion for the partial Riemann hypothesis and we provide some numerical evidence for it using new formulas for the Li coefficients.