The saddle-point method and the Li coefficients

TL;DR

This paper derives a precise asymptotic formula for generalized Li coefficients using the saddle-point method and Nörlund-Rice integrals, under the Generalized Riemann Hypothesis for functions in the Selberg class.

Contribution

It introduces a novel application of the saddle-point method combined with Nörlund-Rice integrals to analyze Li coefficients for functions in the Selberg class.

Findings

Derived an asymptotic formula for Li coefficients under GRH.

Established explicit expressions for constants in the asymptotic formula.

Extended the analysis to a broad class of functions in the Selberg class.

Abstract

In this paper, we apply the saddle-point method in conjunction with the theory of the N$\ddot{o}$rlund-Rice integrals to derive a precise asymptotic formula for the generalized Li coefficients established by Omar and Mazhouda. Actually, for any function $F$ in the Selberg class $\mathcal{S}$ and under the Generalized Riemann Hypothesis, we have $$\lambda_{F}(n)=\frac{d_{F}}{2}n\log n+c_{F}n+O(\sqrt{n}\log n),$$ with $$c_{F}=\frac{d_{F}}{2}(\gamma-1)+\frac{1}{2}\log(\lambda Q_{F}^{2}),\ \ \lambda=\prod_{j=1}^{r}\lambda_{j}^{2\lambda_{j}},$$ where $\gamma$ is the Euler constant and the notation is as bellow.