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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeAug 28th 2014
   • (edited Aug 28th 2014)
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded 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 an elliptic differential operator|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]].

   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:

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   For a complex torus (complex elliptic curve) equipped with its standard flat Riemannian metric, then the zeta function of the corresponding Laplace operator is

   The corresponding functional determinant is

   where is the Dedekind eta function.