The Saddle-Point Method and the Li Coefficients

In this paper, we apply the saddle-point method in conjunction with the theory of the Norlund-Rice integrals to derive precise asymptotic formula for the generalized Li coefficients established by Omar and Mazhouda. Actually, for any function F in the Selberg class S and under the Generalized Riemann Hypothesis, we have lambda(F)(n) = d(F)/2n log n + c(F)n + O(root n log n), with c(F) = d(F)/2 (gamma - 1) + 1/2 log(lambda Q(F)(2)), lambda = Pi(tau)(j=1)lambda(2 lambda j)(j), where gamma is the Euler's constant and the notation is as below.