On asymptotic behavior of generalized Li coefficients in the Selberg class

Abstract

In this paper we obtain a full asymptotic expansion of the archimedean contribution to the Li coefficients λ F (−n) (n is a positive integer) attached to a function F in the certain class S of functions containing the Selberg class S and (unconditionally) the class of all automorphic L-functions attached to irreducible, unitary cuspidal representations of GL N (Q). Applying the obtained results to automorphic L-functions, we improve the result of J.C. Lagarias concerning the asymptotic behavior of archimedean contribution to the nth Li coefficient attached to the automorphic L-function.We also deduce asymptotic behaviors of λ F (−n), as n → +∞ equivalent to Generalized Riemann Hypothesis (GRH) true and GRH false for F ∈ S .

On the modified Li criterion for a certain class of L-functions

“…More precisely, we pose the following two conjectures based on numerical evidence and asymptotic behavior of τ -Li coefficients obtained in [20,17,5].…”

On relations equivalent to the generalized Riemann hypothesis for the Selberg class

“…The importance of the generalized Li coefficients lies in the fact that the GRH for F ∈ S ♯♭ 0 is equivalent to positivity of the sequence of real numbers {Re(λ F (n))} n≥1 . Actually, the GRH for F ∈ S ♯♭ 0 is equivalent to the asymptotic relation (3), as proved in [14] and [16]. Therefore, it is of interest to deduce arithmetic formulas for computation of λ F (n).…”

“…Using the results of [6], [9], [10], [20] and [24]- [27], it is proved in [14,Sec. 4] that the L−function L(s, π) belongs to the class S ♯♭ .…”

“…is called the non-archimedean part of λ F (−n). In [14] and [16] the authors compute the full asymptotic expansion of the archimedean part of the generalized Li coefficient λ F (−n) attached to F ∈ S ♯♭ 0 as n → ∞, and prove that the GRH is equivalent to the asymptotic bound…”

Women in Numbers Europe

“…; T/ C O.1/:(28)It is left to estimate the integral on the left side of the above equation. It is done analogously as inOdžak and Smajlović (2011), using the approximationL.s/ 0 L.s/ D X 1 s C O ; 0 .log jsj/ ;which implies thatL.s/ 0 L.s/ D X jT Im jÄ1 1 s C O ; 0 .log T/ ;…”