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.
$L$-functions, generalized Li coefficients, generalized Riemann hypothesis
"On the asymptotic criterion for the zero-free regions of certain $L$-functions,"
Turkish Journal of Mathematics: Vol. 40:
3, Article 19.
Available at: https://journals.tubitak.gov.tr/math/vol40/iss3/19