FLINT3.7.0-dev
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_arb_poly_zeta_series, arb_poly_zeta_series

Synopsis#include <flint/arb_poly.h>
void _arb_poly_zeta_series(arb_ptr res, arb_srcptr h, slong hlen, const arb_t a, int deflate, slong len, slong prec)source
void arb_poly_zeta_series(arb_poly_t res, const arb_poly_t s, const arb_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, for a constant a>0a > 0, truncated to length n; a=1a = 1 gives the Riemann zeta function. If aa contains a nonpositive number, the result is indeterminate. 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.

The series of ζ(s0+x,a)\zeta(s_0 + x, a) is computed by Euler–Maclaurin summation (_acb_poly_zeta_cpx_series) and composed with s−s0s - s_0 (Zeta functions, polylogarithms and L-functions). For a=1a = 1 and a constant term with negative midpoint the functional equation is used. With s=1+xs = 1 + x and deflate set, the result is ∑k=0n−1(−1)kk!γk(a)xk\sum_{k=0}^{n-1} \frac{(-1)^k}{k!} \gamma_k(a) x^k in terms of the Stieltjes constants γk(a)\gamma_k(a).

Cite this page

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