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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
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