arb_zeta_ui_vec_borwein
#include <flint/arb.h>For efficiency the largest s should be at most about prec; beyond that, the Euler product is better. Arguments near WORD_MAX overflow. For a single s the algorithm is essentially that of MPFR [MPFR2012]: the sum is evaluated backwards to avoid storing more than one coefficient d_k, in integer arithmetic since the terms are slightly larger than 2^{\mathit{prec}}. The only error in the main loop comes from the division by k^s, at most one unit per term; for multi-evaluation the terms are divided repeatedly by k^{\mathit{step}}, and by induction the error per term stays below two units.
References#
- [Bor2000] P. Borwein, "An Efficient Algorithm for the Riemann Zeta Function", Constructive experimental and nonlinear analysis, CMS Conference Proc. 27 (2000) 29-34, http://www.cecm.sfu.ca/personal/pborwein/PAPERS/P155.pdf
- [GS2003] X. Gourdon and P. Sebah, "Numerical evaluation of the Riemann Zeta-function" (2003), http://numbers.computation.free.fr/Constants/Miscellaneous/zetaevaluations.pdf
- [MPFR2012] The MPFR team, "MPFR Algorithms" (2012), https://www.mpfr.org/algo.html
The FLINT team, “arb_zeta_ui_vec_borwein (arb.h),” FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev, 2026. https://flintlib.org/doc/arb/arb_zeta_ui_vec_borwein.html (generated 2026-10-20).
@misc{flint-doc-arb-arb-zeta-ui-vec-borwein,
author = {{The FLINT team}},
title = {{arb\_zeta\_ui\_vec\_borwein (arb.h)}},
howpublished = {FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev},
year = {2026},
url = {https://flintlib.org/doc/arb/arb_zeta_ui_vec_borwein.html},
note = {Generated 2026-10-20}
}