Dedekind eta-function

2020 Mathematics Subject Classification: Primary: 11F20 [MSN][ZBL]

The function defined by

$$\eta(z)=e^{\pi iz/12}\prod_{n=1}^\infty(1-e^{2\pi inz})$$

for $z\in\mathbf C$, $\operatorname{Im}z>0$. As the infinite product converges absolutely, uniformly for $z$ in compact sets (cf. Uniform convergence), the function $\eta$ is holomorphic (cf. Analytic function). Moreover, it satisfies $\eta(z+1)=e^{\pi i/12}\eta(z)$ and $\eta(-1/z)=\sqrt{-iz}\eta(z)$. So, $\eta^{24}$ is a modular form of weight $12$ (cf. also Modular group).

R. Dedekind [a1] comments on computations of B. Riemann in connection with theta-functions (cf. Theta-function). He shows that it is basic to understand the transformation behaviour of the logarithm of the function now carrying his name. This study leads him to quantities now called Dedekind sums (cf. Dedekind sum). See [a2], Chapt. IV, for a further discussion.

References

[a1]	R. Dedekind, "Erläuterungen zu den fragmenten XXVIII" H. Weber (ed.) , B. Riemann: Gesammelte mathematische Werke und wissenschaftlicher Nachlass , Dover, reprint (1953)
[a2]	H. Rademacher, E. Grosswald, "Dedekind sums" , Math. Assoc. America (1972)