View source for Dedekind eta-function

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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|infinite product]] converges absolutely, uniformly for $z$ in compact sets (cf. [[Uniform convergence|Uniform convergence]]), the function $\eta$ is holomorphic (cf. [[Analytic function|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|modular form]] of weight $12$ (cf. also [[Modular group|Modular group]]).

R. Dedekind [[#References|[a1]]] comments on computations of B. Riemann in connection with theta-functions (cf. [[Theta-function|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|Dedekind sum]]). See [[#References|[a2]]], Chapt. IV, for a further discussion.

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Dedekind,   "Erläuterungen zu den fragmenten XXVIII"  H. Weber (ed.) , ''B. Riemann: Gesammelte mathematische Werke und wissenschaftlicher Nachlass'' , Dover, reprint  (1953)</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  H. Rademacher,   E. Grosswald,   "Dedekind sums" , Math. Assoc. America  (1972)</TD></TR>
</table>

[[Category:Number theory]]
[[Category:Special functions]]