FLINT3.7.0-dev
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gr_dedekind_eta, gr_dedekind_eta_q

Synopsis#include <flint/gr_special.h>
int gr_dedekind_eta(gr_ptr res, gr_srcptr tau, gr_ctx_t ctx)source
int gr_dedekind_eta_q(gr_ptr res, gr_srcptr q, gr_ctx_t ctx)source

Set res to the Dedekind eta function η(τ)\eta(\tau). gr_dedekind_eta_q takes the nome q=e2πiτq = e^{2 \pi i \tau} instead and computes η\eta without the leading factor q1/24q^{1/24}, that is, q−1/24η(τ)=∏k≥1(1−qk)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 qq for ∣q∣<1|q| < 1. It returns GR_DOMAIN if ∣q∣≥1|q| \ge 1, and is implemented for real and complex balls.

For ∣q∣≤1/2|q| \le 1/2 (or a ball qq containing 0) the ball versions sum the series with acb_modular_eta_sum; otherwise they compute τ=log⁡(q)/(2πi)\tau = \log(q) / (2 \pi i) and use acb_modular_eta, which moves τ\tau to the fundamental domain, and multiply by q−1/24=e−log⁡(q)/24q^{-1/24} = e^{-\log(q)/24}. A ball qq that contains points of modulus both below and at least 1 gives GR_UNABLE.

Cite this page

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 2026-11-02).