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On the asymptotic criterion for the zero-free regions of certain \(L\)-functions. (English) Zbl 1424.11128
Summary: We investigate relations between zero-free regions of certain \(L\)-functions and the asymptotic behavior of corresponding generalized Li coefficients. Precisely, we prove that violation of the \(\tau/2\)-generalized Riemann hypothesis implies oscillations of corresponding \(\tau\)-Li coefficients with exponentially growing amplitudes. Results are obtained for class \(\mathcal{S}^{\natural \flat}(\sigma_0, \sigma_1)\) that contains the Selberg class, the class of all automorphic \(L\)-functions, the Rankin-Selberg \(L\)-functions, and products of suitable shifts of the mentioned functions.
MSC:
11M26 Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses
11M41 Other Dirichlet series and zeta functions
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