Reformulation of the Li criterion for the Selberg class

Abstract

Let $F$ be a function in the Selberg class $\mathcal{S}$ and $a$ be a real number not equal to 1/2. Consider the sum $${\lambda }_F(n,a)=\sum _{\rho }\left[1-{\left(\frac{\rho -a}{\rho +a-1}\right)}^n\right],$$ where $\rho$ 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" ${\lambda }_F(n,a)$, for $n=1,2,..$ and $a<1/2$. Furthermore, we give an explicit arithmetic and asymptotic formula of these coefficients.

32 pages

References in corpus (3)

• On certain sums over the nontrivial zeta zeros
• Generalized Bombieri - Lagarias' theorem and generalized Li's criterion
• An arithmetic interpretation of generalized Li's criterion