EXPLICIT FORMULAS FOR DIRICHLET AND HECKE L-FUNCTIONS

摘要

In 1997, the author proved that the Riemann hypothesis holds if and only if λ_n = ∑ [1 - (1 - 1/ρ)~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 λ_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.