The saddle-point method and the Li coefficients

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.

Add explanations, especially for formula (4.4)

References in corpus (2)

• On Differences of Zeta Values
• A sharpening of Li's criterion for the Riemann Hypothesis

Cited by in corpus (1)

• New asymptotic formulas for sums over zeros of functions from the Selberg class