The Saddle-Point Method and the Li Coefficients

Abstract

Abstract. In this paper, we apply the saddle-point method in conjunction with the theory of the Nörlund-Rice integrals to derive precise asymptotic formula for the generalized Li coefficients established by Omar and Mazhouda.

Correction to: New asymptotic formulas for sums over zeros of functions from the Selberg class, Ann. Sci. Math. Québec 34 (2010), no 1, 85–108

“…Using the corollary in [2, Theorem 1], we get the following generalization of the Li criterion for the Riemann hypothesis. In a recent works [10] and [17], new asymptotic formula was given for the generalized Li coefficients. To do so, we used two different methods, both of them give the same main term.…”

“…[10] [17] Let F ∈ S. ThenRH ⇔ λ F (−n) = d F 2 n log n + c F n + O( √ n log n),and γ 0 is the Euler constant.…”

“…In a recent works [10] and [17], new asymptotic formula was given for the generalized Li coefficients. To do so, we used two different methods, both of them give the same main term.…”

Reformulation of the Li criterion for the Selberg class

“…Theorem 3 was proved also in [11] using the saddle-point method in conjunction with the theory of the Nörlund-Rice integrals.…”

“…In this section, we use the same argument as in [11,Theorem 4.1] which use the saddle point method in conjunction with the Nörlund-Rice integral to deduce an asymptotic formula for the modified Li coefficients equivalent to the Riemann hypothesis.…”