gr_dedekind_eta, gr_dedekind_eta_q
#include <flint/gr_special.h>Set res to the Dedekind eta function \eta(\tau). gr_dedekind_eta_q takes the nome q = e^{2 \pi i \tau} instead and computes \eta without the leading factor q^{1/24}, that is, q^{-1/24} \eta(\tau) = \prod_{k \ge 1} (1 - q^k) (as acb_modular_eta_sum does), which is a single-valued function of q for |q| < 1. It returns GR_DOMAIN if |q| \ge 1, and is implemented for real and complex balls.
For |q| \le 1/2 (or a ball q containing 0) the ball versions sum the series with acb_modular_eta_sum; otherwise they compute \tau = \log(q) / (2 \pi i) and use acb_modular_eta, which moves \tau to the fundamental domain, and multiply by q^{-1/24} = e^{-\log(q)/24}. A ball q that contains points of modulus both below and at least 1 gives GR_UNABLE.
The FLINT team, “gr_dedekind_eta, gr_dedekind_eta_q (gr_special.h),” FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev, 2026. https://flintlib.org/doc/gr_special/gr_dedekind_eta.html (generated 2026-10-20).
@misc{flint-doc-gr-special-gr-dedekind-eta,
author = {{The FLINT team}},
title = {{gr\_dedekind\_eta, gr\_dedekind\_eta\_q (gr\_special.h)}},
howpublished = {FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev},
year = {2026},
url = {https://flintlib.org/doc/gr_special/gr_dedekind_eta.html},
note = {Generated 2026-10-20}
}