`acb_modular_eta_sum`

Synopsis#include <flint/acb_modular.h>

void acb_modular_eta_sum(acb_t eta, const acb_t q, slong prec)source

Sets eta to $\prod_{n \ge 1} (1 - q^n) = \sum_{k \in \mathbb{Z}} (-1)^k q^{k(3k-1)/2}$ , the eta function without the factor $e^{\pi i \tau / 12}$ , given $q = e^{2\pi i \tau}$ , by summing the pentagonal number series with a rigorous tail bound. Like acb_modular_theta_sum, it is meant for $|q| \ll 1$ . The series is summed with a short addition sequence, or with rectangular splitting from 400 terms [EHJ2016].

References#

[EHJ2016] A. Enge, W. Hart and F. Johansson, "Short addition sequences for theta functions", preprint (2016), https://arxiv.org/abs/1608.06810

Cite this page

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

@misc{flint-doc-acb-modular-acb-modular-eta-sum,
  author       = {{The FLINT team}},
  title        = {{acb\_modular\_eta\_sum (acb\_modular.h)}},
  howpublished = {FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev},
  year         = {2026},
  url          = {https://flintlib.org/doc/acb_modular/acb_modular_eta_sum.html},
  note         = {Generated 2026-11-02}
}

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