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.