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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:

   ---

   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.