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On asymptotic behavior of generalized Li coefficients. (English) Zbl 1429.11172
Summary: In this paper, we consider the asymptotic behaviour of \(\tau\)-Li coefficients for the wide class of \(L\)-functions that contains the Selberg class, the class of all automorphic \(L\)-functions, the Rankin-Selberg \(L\)-functions, as well as products of suitable shifts of the mentioned functions. We consider both archimedean and non-archimedean contribution to the \(\tau\)-Li coefficients, both separately, and their joint contribution to the coefficients. We also derive the behavior of the coefficients in the case the \(\tau/2\)-Riemann hypothesis holds, which is the generalization of the Riemann hypothesis for the class under consideration. Finally, we conclude with some examples and numerics.
MSC:
11M41 Other Dirichlet series and zeta functions
11M26 Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses
Software:
Arb
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References:
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