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# The Saddle-Point Method and the Li Coefficients

Published:2011-02-10
Printed: Jun 2011
• Kamel Mazhouda,
Faculté des sciences de Monastir, Département de mathématiques, 5000 Monastir, Tunisia
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## Abstract

In this paper, we apply the saddle-point method in conjunction with the theory of the Nörlund-Rice integrals to derive 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's constant and the notation is as below.
 Keywords: Selberg class, Saddle-point method, Riemann Hypothesis, Li's criterion
 MSC Classifications: 11M41 - Other Dirichlet series and zeta functions {For local and global ground fields, see 11R42, 11R52, 11S40, 11S45; for algebro-geometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72} 11M06 - $\zeta (s)$ and $L(s, \chi)$