The Saddle-Point Method and the Li Coefficients

Mazhouda, Kamel

Canadian mathematical bulletin, Tome 54 (2011) no. 2, pp. 316-329

Résumé

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.

Détail

DOI : 10.4153/CMB-2011-016-6

Mots-clés : 11M41, 11M06, Selberg class, Saddle-point method, Riemann Hypothesis, Li's criterion