@misc{dedekindeta2025,
  shorthand    = {DED},
  author       = {Koustav Mondal},
  title        = {The {Dedekind Eta Database}},
  howpublished = {\url{https://kosutav1997.github.io/Dedekind-Eta-Database/}},
  year         = {2025},
  note         = {[Online; accessed ]},
}

Dedekind Eta Database

Introduction

The Dedekind eta function, \( \eta(\tau) \), is a modular form with the following properties:

• Holomorphic on the upper half-plane: \( \operatorname{Im}(\tau) > 0 \).
• q-expansion:
  \[ \eta(\tau) = q^{1/24} \prod_{n=1}^{\infty} (1 - q^n), \quad q = e^{2 \pi i \tau} \]

  From (1), it is evident that,

  \[ \eta(iy) \in \mathbb{R}^{+}, \quad e^{\frac{-i\pi}{24}}\eta\left(\frac{1+iy}{2}\right) \in \mathbb{R}^{+}, \quad y>0. \]
• Modular transformations (examples):

\[ \eta(\tau + 1) = e^{\pi i /12} \eta(\tau) \]

\[ \eta(-1/\tau) = \sqrt{-i\tau} \, \eta(\tau) \]

If we choose \( u(\tau) = \left( \frac{\eta(2\tau)}{\eta(\tau)} \right)^{24} \) to be a Hauptmodul for \( \Gamma_0(2) \), then it is related to the j-invariant as:

\[ j(\tau) = \frac{(1 + 256 \cdot u(\tau))^3}{u(\tau)} \]

• Work of Muzaffar and Williams ^{[1], [2]}

Let \(\lambda \in \mathbb{N}, d=-4\lambda\), such that \(d\equiv 0 \text{ or } 1\pmod{4}\). Choose largest \(f\) such that \(f^2\mid d, d/f^2\equiv 0 \text{ or } 1\pmod{4}\). Set \(\Delta=d/f^2\). Let \(v_p(f)\) be a non-negative integer such that \(p^{v_p(f)}\mid p\) but \(p^{v_p(f)+1}\nmid p\). Then, define

\[ \alpha_p(\Delta, f)=\frac{(p^{v_p(f)}-1)\left(1-\left(\frac{\Delta}{p}\right) \right)}{ p^{v_p(f)-1}(p-1) \left(p - \left(\frac{\Delta}{p}\right)\right)} \]

where \( \left(\frac{\Delta}{p}\right)\) is the Legendre symbol. Let \(w(d)\) denote the number of automorphs of primitive, positive-definite, binary quadratic form of discriminant \(d\). Then, \(w(d)=\begin{cases} 6, &\text{ if } d=-3\\ 4, &\text{ if } d=-4\\ 2, &\text{ if } d<-4 \end{cases}\).

The positive-definite, primitive, integral, binary quadratic form \(ax^2+bxy+cy^2\), denoted by \((a,b,c)\), with discriminant \(d=b^2-4ac\), forms a finite abelian group under Gaussian composition, denoted by \(H(d)\) and the order of \(H(d)\) is denoted by \(h(d)\).

\[ \eta(\sqrt{-\lambda})=2^{-3/4}\pi^{-1/4}\lambda^{-1/4}\prod_{p\mid f}p^{\alpha_p(\Delta,f)/4}\left( \prod_{m=1}^{|\Delta|}\Gamma\left(\frac{m}{|\Delta|}\right)^{\left(\frac{\Delta}{m}\right)} \right)^{\frac{w(\Delta)}{8h(\Delta)}}e^{-E(K,d)} \]