Rocky Mountain Journal of Mathematics

On the $\tau $-Li coefficients for automorphic $L$-functions

Kamel Mazhouda

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Abstract

In this paper, we extend the Li coefficients for automorphic $L$-functions and the Li criterion for the Riemann hypothesis to yield a necessary and sufficient condition for the existence of zero-free strips for automorphic $L$-functions inside the critical strip. Next, we give an arithmetical and asymptotical formula for these coefficients. Finally, we show that there exists an entire function of exponential type that interpolates the extended Li coefficients (or the $\tau $-Li coefficients) at integer values. The results of this paper arise from ideas of the author~\cite {15}, Freitas~\cite {8}, Lagarias~\cite {10} and Odz$\breve {a}$k and Smajlovi$\grave {c}$~\cite {17}.

Article information

Source
Rocky Mountain J. Math., Volume 47, Number 6 (2017), 1987-2011.

Dates
First available in Project Euclid: 21 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1511254961

Digital Object Identifier
doi:10.1216/RMJ-2017-47-6-1987

Mathematical Reviews number (MathSciNet)
MR3725253

Zentralblatt MATH identifier
06816579

Subjects
Primary: 11M06: $\zeta (s)$ and $L(s, \chi)$ 11M36: Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. Explicit formulas 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}

Keywords
Automorphic $L$-functions Dirichlet series Li's criterion Riemann hypothesis

Citation

Mazhouda, Kamel. On the $\tau $-Li coefficients for automorphic $L$-functions. Rocky Mountain J. Math. 47 (2017), no. 6, 1987--2011. doi:10.1216/RMJ-2017-47-6-1987. https://projecteuclid.org/euclid.rmjm/1511254961


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