A note on Li’s coefficients for Hecke L-functions

Abstract

Recently, the Li criterion for the Riemann hypothesis has been extended for a general class of [Formula: see text]-functions, so-called the Selberg class [S. Omar and K. Mazhouda, Le critère de Li et l’hypothèse de Riemann pour la classe de Selberg, J. Number Theory 125(1) (2007) 50–58; Corrigendum et addendum à “Le critère de Li et l’hypothèse de Riemann pour la classe de Selberg” [J. Number Theory 125(1) (2007) 50–58], J. Number Theory 130(4) (2010) 1109–1114]. Further numerical computations have been done to verify the positivity of some Li coefficients for the Dirichlet [Formula: see text]-functions and the Hecke [Formula: see text]-functions [S. Omar, R. Ouni and K. Mazhouda, On the zeros of Dirichlet [Formula: see text]-functions, LMS J. Comput. Math. 14 (2011) 140–154; On the Li coefficients for the Hecke [Formula: see text]-functions, Math. Phys. Anal. Geom. 17(1–2) (2014) 67–81]. Based on the latter numerical experiments, it was conjectured that those coefficients are increasing in [Formula: see text]. In this note, we show actually that the Riemann hypothesis holds if and only if the Li coefficients for the Hecke [Formula: see text]-functions are increasing in [Formula: see text].