An effective asymptotic formula for the Stieltjes constants

New results on the Stieltjes constants: Asymptotic and exact evaluation

The Stieltjes constants γk(a) are the expansion coefficients in the Laurent series for the Hurwitz zeta function about s = 1. We present new asymptotic, summatory, and other exact expressions for these and related constants.

AN ASYMPTOTIC FORM FOR THE STIELTJES CONSTANTS γk(a) AND FOR A SUM Sγ(n) APPEARING UNDER THE LI CRITERION

We present several asymptotic analyses for quantities associated with the Riemann and Hurwitz zeta functions. We first determine the leading asymptotic behavior of the Stieltjes constants γk(a). These constants appear in the regular part of the Laurent expansion of the Hurwitz zeta function. We then use asymptotic results for the Laguerre polynomials Ln to investigate a certain sum Sγ(n) involv...

Series representations for the Stieltjes constants

The Stieltjes constants γk(a) appear as the coefficients in the regular part of the Laurent expansion of the Hurwitz zeta function ζ(s, a) about s = 1. We present series representations of these constants of interest to theoretical and computational analytic number theory. A particular result gives an addition formula for the Stieltjes constants. As a byproduct, expressions for derivatives of a...

An Explicit Formula Relating Stieltjes Constants and Li’s Numbers

In this paper we present a new formula relating Stieltjes numbers γn and Laurent coefficinets ηn of logarithmic derivative of the Riemann’s zeta function. Using it we derive an explicit formula for the oscillating part of Li’s numbers ∼ λn which are connected with the Riemann hypothesis.

Asymptotic estimates for Stieltjes constants: a probabilistic approach