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The Saddle-Point Method and the Li Coefficients

Published online by Cambridge University Press:  20 November 2018

Kamel Mazhouda*
Affiliation:
Faculté des sciences de Monastir, Département de mathématiques, 5000 Monastir, Tunisia e-mail: kamel.mazhouda@fsm.rnu.tn
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Abstract

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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

$${{c}_{F}}\,=\,\frac{{{d}_{F}}}{2}(\gamma \,-\,1)\,+\,\frac{1}{2}\log (\lambda \text{Q}_{F}^{2}),\,\,\lambda \,=\,\prod\limits_{j=1}^{r}{\lambda _{j}^{2{{\lambda }_{j}}}},$$

where $\gamma $ is the Euler's constant and the notation is as below.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2011

References

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