An asymptotic expansion for the Stieltjes constants

2015/08/17 by R. B. Paris, Paris, R. B.
Mathematics · #11M06 #30E20 #34E05 #41A60 #Advanced Mathematical Identities #Analytic Number Theory Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical functions and polynomials

paper · pdf · doi:10.48550/arxiv.1508.03948

openalex publication_date 2015/08/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The Stieltjes constants γ_n appear in the coefficients in the Laurent expansion of the Riemann zeta function ζ(s) about the simple pole s=1. We present an asymptotic expansion for γ_n as n→ ∞ 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.

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