FLINT3.7.0-dev
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_acb_poly_zeta_series, acb_poly_zeta_series

Synopsis#include <flint/acb_poly.h>
void _acb_poly_zeta_series(acb_ptr res, acb_srcptr h, slong hlen, const acb_t a, int deflate, slong len, slong prec)source
void acb_poly_zeta_series(acb_poly_t res, const acb_poly_t f, const acb_t a, int deflate, slong n, slong prec)source

Set res to the Hurwitz zeta function ζ(s,a)\zeta(s, a) of the power series ss = f for a constant aa, truncated to length n; a=1a = 1 gives the Riemann zeta function. If deflate is nonzero, ζ(s,a)+1/(1−s)\zeta(s, a) + 1/(1 - s) is computed, which is finite at s=1s = 1; with s=1+xs = 1 + x this gives the Stieltjes constants, ∑k=0n−1(−1)kk!γk(a)xk\sum_{k=0}^{n-1} \frac{(-1)^k}{k!} \gamma_k(a) x^k.

The series at the constant term is computed by acb_dirichlet_zeta_jet if a=1a = 1 and by _acb_poly_zeta_cpx_series otherwise, and then composed with the nonconstant part of ss. acb_dirichlet_zeta_jet uses the functional equation if the real part of the midpoint is negative, the Riemann–Siegel formula for at most two coefficients at very large imaginary parts (above about 60 prec3/260\,\mathit{prec}^{3/2}), and Euler–Maclaurin summation otherwise.

Guides#

Cite this page

The FLINT team, “_acb_poly_zeta_series, acb_poly_zeta_series (acb_poly.h),” FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev, 2026. https://flintlib.org/doc/acb_poly/_acb_poly_zeta_series.html (generated 2026-11-02).