On the zeros of Dirichlet $L$-functions

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.

Proposition 1 page 143 is revised