- Turkish Journal of Mathematics
- Vol: 40 Issue: 3
- On the asymptotic criterion for the zero-free regions of certain $L$-functions
On the asymptotic criterion for the zero-free regions of certain $L$-functions
Authors : Almasa Odzak
Pages : 688-702
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Publication Date : 9999-12-31
Article Type : Makaleler
Abstract :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 $\shfs$ 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.Keywords : $L$-functions, generalized Li coefficients, generalized Riemann hypothesis