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March 2017 On relations equivalent to the generalized Riemann hypothesis for the Selberg class
Kamel Mazhouda, Lejla Smajlović
Funct. Approx. Comment. Math. 56(1): 67-93 (March 2017). DOI: 10.7169/facm/1593

Abstract

We prove that the generalized Riemann hypothesis (GRH) for functions in the class $\mathcal{S}^{\sharp\flat}$ containing the Selberg class is equivalent to a certain integral expression of the real part of the generalized Li coefficient $\lambda_F(n)$ associated to $F\in\mathcal{S}^{\sharp\flat}$, for positive integers $n$. Moreover, we deduce that the GRH is equivalent to a certain expression of $\Re(\lambda_F(n))$ in terms of the sum of the Chebyshev polynomials of the first kind. Then, we partially evaluate the integral expression and deduce further relations equivalent to the GRH involving the generalized Euler-Stieltjes constants of the second kind associated to $F$. The class $\mathcal{S}^{\sharp\flat}$ unconditionally contains all automorphic $L$-functions attached to irreducible cuspidal unitary representations of $\mathrm{GL}_N(\mathbb{Q})$, hence, as a corollary we also derive relations equivalent to the GRH for automorphic $L$-functions.

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Kamel Mazhouda. Lejla Smajlović. "On relations equivalent to the generalized Riemann hypothesis for the Selberg class." Funct. Approx. Comment. Math. 56 (1) 67 - 93, March 2017. https://doi.org/10.7169/facm/1593

Information

Published: March 2017
First available in Project Euclid: 27 January 2017

zbMATH: 06864147
MathSciNet: MR3629012
Digital Object Identifier: 10.7169/facm/1593

Subjects:
Primary: 11M06
Secondary: 11M26, 11M36, 11M41

Rights: Copyright © 2017 Adam Mickiewicz University

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Vol.56 • No. 1 • March 2017
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