An asymptotic expansion for the Stieltjes constants

TL;DR

This paper derives an asymptotic expansion for the Stieltjes constants, providing explicit formulas and numerical evidence of its accuracy for large n, advancing understanding of these constants in analytic number theory.

Contribution

It introduces a new asymptotic expansion for the Stieltjes constants with explicit coefficients and demonstrates its effectiveness through numerical validation.

Findings

The expansion accurately approximates $\gamma_n$ for large n.

Explicit coefficients are provided for practical computation.

Numerical results confirm the expansion's high accuracy.

Abstract

The Stieltjes constants $\gamma_n$ appear in the coefficients in the Laurent expansion of the Riemann zeta function $\zeta(s)$ about the simple pole $s=1$. We present an asymptotic expansion for $\gamma_n$ as $n\rightarrow \infty$ based on the approach described by Knessel and Coffey [Math. Comput. {\bf 80} (2011) 379--386]. A truncated form of this expansion with explicit coefficients is also given. Numerical results are presented that illustrate the accuracy achievable with our expansion.