acb_dirichlet_secondary_zeta
Synopsis
#include <flint/acb_dirichlet.h>Sets res to the secondary zeta function Z(s) = \sum_{n=1}^{\infty} \gamma_n^{-s}, where \rho_n = 1/2 + i\gamma_n runs over the zeros of \zeta(s) with positive imaginary part. The Riemann hypothesis is not assumed. The algorithm follows [Ari2020], with the decomposition Z(s) = A(s) - P(s) + E(s) - S(s) depending on a parameter a > 0, an improved tail bound for P, and rigorous bounds for the tails of A and S that replace the heuristic bounds of [Ari2020].
References#
- [Ari2020] J. Arias de Reyna, "Computation of the secondary zeta function", https://arxiv.org/abs/2006.04869
Cite this page
The FLINT team, “acb_dirichlet_secondary_zeta (acb_dirichlet.h),” FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev, 2026. https://flintlib.org/doc/acb_dirichlet/acb_dirichlet_secondary_zeta.html (generated 2026-10-20).
@misc{flint-doc-acb-dirichlet-acb-dirichlet-secondary-zeta,
author = {{The FLINT team}},
title = {{acb\_dirichlet\_secondary\_zeta (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_secondary_zeta.html},
note = {Generated 2026-10-20}
}