3
8. 4. 2016.
On the asymptotic criterion for the zero-free regions of certain $L$-functions
: 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 (cid:28)= 2-generalized Riemann hypothesis implies oscillations of corresponding (cid:28) -Li coefficients with exponentially growing amplitudes. Results are obtained for class S ♯♭ ( (cid:27) 0 ; (cid:27) 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.