Dedekind eta function

Definition: Let \(q=e^{2 \pi iz}\), the *Dedekind eta function* is defined as:

$$
\eta(z)=q^{\frac{1}{24}}\prod_{n=1}^{\infty}(1-q^{n})\quad z\in\mathbb H.
$$

Properties:
  • Holomorphic and non-vanishing on the upper half-plane \(\mathbb H.\)
  • weakly modular form of weight 1/2.
  • Periodic: \(\eta(z+24)=\eta(z).\)
  • \(\eta(z+1)=e^{\pi i/12}\eta{z}\)
  • Modularity: \(\eta\left(-\frac{1}{z}\right)=\sqrt{-iz}\eta(z)\)