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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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##### References:
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