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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeAug 28th 2014
• (edited Aug 28th 2014)

added the following story to the Properties-section of Dedekind eta function and also to the Examples-section of functional determinant and zeta function of an elliptic differential operator:

For $E = \mathbb{C}/(\mathbb{Z}\oplus \tau \mathbb{Z})$ a complex torus (complex elliptic curve) equipped with its standard flat Riemannian metric, then the zeta function of the corresponding Laplace operator $\Delta$ is

$\zeta_{\Delta} = (2\pi)^{-2 s} E(s) \coloneqq (2\pi)^{-2 s} \underset{(k,l)\in \mathbb{Z}\times\mathbb{Z}-(0,0)}{\sum} \frac{1}{{\vert k +\tau l\vert}^{2s}} \,.$

The corresponding functional determinant is

$\exp( E^\prime(0) ) = (Im \tau)^2 {\vert \eta(\tau)\vert}^4 \,,$

where $\eta$ is the Dedekind eta function.