`acb_dirichlet_hurwitz_precomp_init`

Synopsis#include <flint/acb_dirichlet.h>

void acb_dirichlet_hurwitz_precomp_init(acb_dirichlet_hurwitz_precomp_t pre, const acb_t s, int deflate, slong A, slong K, slong N, slong prec)

source

Precomputes a grid of $N$ Taylor polynomials of length $K$ for $\zeta(s, a + A)$ , for fast evaluation of $\zeta(s, a)$ at $a \in (0,1]$ with fixed $s$ . This costs $NK$ Hurwitz zeta values; each later evaluation costs $2K$ arithmetic operations plus $A$ powers, with a truncation error $O(1/(2AN)^K)$ . With deflate nonzero, the deflated function (without the pole at $s = 1$ ) is used.

With $A = 0$ , no precomputation is done and evaluation is the same as calling acb_dirichlet_hurwitz. Otherwise $A$ , $K$ and $N$ must be positive (an exception is thrown otherwise), and the error bound is finite only if $K + \operatorname{Re}(s) > 1$ . To avoid an initial bump that loses precision and slows convergence, $AN$ should be at least about as large as $|s|$ , for instance $AN > 0.5 |s|$ . The error bound is that of Taylor expansion in the parameter.

Cite this page

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

@misc{flint-doc-acb-dirichlet-acb-dirichlet-hurwitz-precomp-init,
  author       = {{The FLINT team}},
  title        = {{acb\_dirichlet\_hurwitz\_precomp\_init (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_hurwitz_precomp_init.html},
  note         = {Generated 2026-11-02}
}

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