Reformulation of the Li criterion for the Selberg class

TL;DR

This paper extends Li's criterion for the Riemann hypothesis to functions in the Selberg class by introducing a modified coefficient and proving its positivity is equivalent to the hypothesis, along with explicit formulas for these coefficients.

Contribution

It reformulates Li's criterion for the Selberg class using a modified coefficient, establishing its positivity as equivalent to the Riemann hypothesis and providing explicit formulas.

Findings

Positivity of modified Li coefficients is equivalent to the Riemann hypothesis.

Derived explicit arithmetic and asymptotic formulas for the coefficients.

Extended Li's criterion to the broader 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.