Explicit formulas for Dirichlet and Hecke $L$-functions

Abstract

In 1997, the author proved that the Riemann hypothesis holds if and only if $\lambda_n=\sum [1-(1-1/\rho)^n]>0$ for all positive integers $n$, where the sum is over all complex zeros of the Riemann zeta function. In 1999, E. Bombieri and J. Lagarias generalized this result and obtained a remarkable general theorem about the location of zeros. They also gave an arithmetic interpretation for the numbers $\lambda_n$. In this note, the author extends Bombieri and Lagarias' arithmetic formula to Dirichlet $L$-functions and to $L$-series of elliptic curves over rational numbers.