On the $\tau $-Li coefficients for automorphic $L$-functions

Abstract

In this paper, we extend the Li coefficients for automorphic $L$-functions and the Li criterion for the Riemann hypothesis to yield a necessary and sufficient condition for the existence of zero-free strips for automorphic $L$-functions inside the critical strip. Next, we give an arithmetical and asymptotical formula for these coefficients. Finally, we show that there exists an entire function of exponential type that interpolates the extended Li coefficients (or the $\tau $-Li coefficients) at integer values. The results of this paper arise from ideas of the author~\cite {15}, Freitas~\cite {8}, Lagarias~\cite {10} and Odz$\breve {a}$k and Smajlovi$\grave {c}$~\cite {17}.