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On asymptotic behavior of generalized Li coefficients in the Selberg class. (English) Zbl 1257.11082
Summary: In this paper we obtain a full asymptotic expansion of the archimedean contribution to the Li coefficients $$\lambda F(-n)$$ ($$n$$ is a positive integer) attached to a function $$F$$ in the certain class $$S'$$ of functions containing the Selberg class $$S$$ and (unconditionally) the class of all automorphic $$L$$-functions attached to irreducible, unitary cuspidal representations of $$\text{GL}_N(\mathbb Q)$$. Applying the obtained results to automorphic $$L$$-functions, we improve the result of J. C. Lagarias concerning the asymptotic behavior of archimedean contribution to the $$n$$th Li coefficient attached to the automorphic $$L$$-function. We also deduce the asymptotic behavior of $$\lambda F(-n)$$ as $$n\to+\infty$$ equivalent to Generalized Riemann Hypothesis (GRH) true and GRH false for $$F\in S'$$.

##### MSC:
 11M41 Other Dirichlet series and zeta functions 11M26 Nonreal zeros of $$\zeta (s)$$ and $$L(s, \chi)$$; Riemann and other hypotheses 11S40 Zeta functions and $$L$$-functions
##### Keywords:
Selberg class; Li’s coefficients
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##### References:
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