On the asymptotic criterion for the zero-free regions of certain LL-functions

Abstract

We investigate relations between zero-free regions of certain L-functions and the asymptotic behavior of corresponding generalized Li coefficients. Precisely, we prove that violation of the τ /2 -generalized Riemann hypothesis implies oscillations of corresponding τ -Li coefficients with exponentially growing amplitudes. Results are obtained for class S ♯♭(σ0, σ1) that contains the Selberg class, the class of all automorphic L-functions, the Rankin Selberg Lfunctions, and products of suitable shifts of the mentioned functions.