Reformulation of the Li criterion for the Selberg class

Abstract

Let F be a function in the Selberg class S and a be a real number not equal to 1/2. Consider the sumwhere ρ runs over the non-trivial zeros of F . In this paper, we prove that the Riemann hypothesis is equivalent to the positivity of the "modified Li coefficient" λ F (n, a), for n = 1, 2, .. and a < 1/2. Furthermore, we give an explicit arithmetic and asymptotic formula of these coefficients.