Rocky Mountain Journal of Mathematics

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

Kamel Mazhouda

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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}.

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

First available in Project Euclid: 21 November 2017

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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}

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


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.

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