FLINT3.7.0-dev

acb_dirichlet_zeta_jet

Synopsis#include <flint/acb_dirichlet.h>
void acb_dirichlet_zeta_jet(acb_t res, const acb_t s, int deflate, slong len, slong prec)source

Sets res (a vector of length len) to the Taylor coefficients of ζ(s+x)\zeta(s + x), or of the deflated function ζ(s+x)−1/(s+x−1)\zeta(s + x) - 1/(s + x - 1) if deflate is nonzero. Euler–Maclaurin summation is used (after the reflection formula when Re⁡(s)≤−4\operatorname{Re}(s) \le -4, or ≤−2.5\le -2.5 at precision below 128 bits; for larger negative Re⁡(s)\operatorname{Re}(s) it is cheaper and loses fewer bits to sum directly at ss, with guard bits proportional to −Re⁡(s)log⁡2p-\operatorname{Re}(s) \log_2 p), the Riemann–Siegel formula for len ≤2\le 2 high on the critical line, or, for an exact rational number ss of small height (including integers) at very high precision, the approximate functional equation (acb_dirichlet_l_fmpq_afe_jet), whose cost is O~(p1.5)\tilde O(p^{1.5}) instead of O~(p2)\tilde O(p^2) and which needs far less memory.

Guides#

Cite this page

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