acb_dirichlet_stieltjes
#include <flint/acb_dirichlet.h>Sets res to the generalised Stieltjes constant \gamma_n(a) for an integer n \ge 0 (an exception is thrown for n < 0), the coefficient in the Laurent series \zeta(s,a) = \frac{1}{s-1} + \sum_{n=0}^\infty \frac{(-1)^n}{n!} \gamma_n(a) (s-1)^n. With a = 1 these are the ordinary Stieltjes constants.
The function uses an integral representation that allows fast computation for extremely large n [JB2018]. If n is moderate and the precision high enough, it instead expands the Hurwitz zeta function in a power series at s = 1 and reads off the last coefficient. To compute all of \gamma_0(a), \ldots, \gamma_n(a), expanding \zeta(s, a) once (for instance with acb_poly_zeta_series) is faster than calling this function repeatedly, unless n is at least several hundred.
References#
- [JB2018] F. Johansson and I. Blagouchine. "Computing Stieltjes constants using complex integration", preprint (2018), https://arxiv.org/abs/1804.01679
The FLINT team, “acb_dirichlet_stieltjes (acb_dirichlet.h),” FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev, 2026. https://flintlib.org/doc/acb_dirichlet/acb_dirichlet_stieltjes.html (generated 2026-10-20).
@misc{flint-doc-acb-dirichlet-acb-dirichlet-stieltjes,
author = {{The FLINT team}},
title = {{acb\_dirichlet\_stieltjes (acb\_dirichlet.h)}},
howpublished = {FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev},
year = {2026},
url = {https://flintlib.org/doc/acb_dirichlet/acb_dirichlet_stieltjes.html},
note = {Generated 2026-10-20}
}