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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 Selberg class, Saddle-point method, Riemann Hypothesis, Li's criterion
MSC Classifications: 11M41, 11M06 show english descriptions 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}
$\zeta (s)$ and $L(s, \chi)$
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 $L(s, \chi)$
 

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