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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=/(τ)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 Δ\Delta is

    ζ Δ=(2π) 2sE(s)(2π) 2s(k,l)×(0,0)1|k+τl| 2s. \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 (0))=(Imτ) 2|η(τ)| 4, \exp( E^\prime(0) ) = (Im \tau)^2 {\vert \eta(\tau)\vert}^4 \,,

    where η\eta is the Dedekind eta function.

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