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Is there any way to regularize this infinite product?

$$ \det \Delta = \prod_{a,b > 0} (a^2 + b^2) $$

I tried to say this was the derivative at $0$ of an L-function

$$ \zeta_{K}'(0) = \sum_{(a,b)} \log (a^2 + b^2) $$

and perhaps this can be evaluated using a functional equation (which I don't know off-hand).

I also said that on the torus $S^1 \times S^1$ as eigenvalues $a^2 + b^2$ on the basis (of characters) $e(a^2 + b^2) = e^{2\pi i(a^2 + b^2)}$.

$$ \left( \frac{\partial^2}{\partial x^2}+ \frac{\partial^2}{\partial y^2}\right) e^{2\pi i \, (ax + by)} = (a^2 + b^2 )e^{2\pi i \, (ax + by)} $$

Then I said we could write the Laplaian operator in the Fourrier basis:

$$ \Delta = \sum_{(a,b) \in \mathbb{Z}^2} \big( a^2 + b^2 \big) e^{2\pi i \, (ax + by)} $$

where hopefully this could make sense using theory of distributions.

This was motivated by trying to understand the Gross-Zagier formula, but I wound up using none of it.

Also I found it odd to see the same regularization formulas over and over in physics papers, but sometimes, we can regulaize an L-function but there is no differential operator to go with it.

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  • $\begingroup$ @reuns I'm trying to write in a style that's typical of what I'm seeing in the hep-th section of arXiv. I'm asking what happens if you set $s=0$ in the zeta functions, and what happens if $z \in \mathbb{R}$ for the theta functions. With the formulas you've provided we can regularize the products in question. $\endgroup$
    – john mangual
    Commented Nov 18, 2017 at 21:58
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    $\begingroup$ Your question is unclear. $ 4 \zeta(s) L(s,\chi_4) = 4\zeta_{\mathbb{Q}(i)}(s) = \sum_{(a,b) \in \mathbb{Z}^2 \setminus (0,0)} (a^2+b^2)^{-s} = \frac{1}{(2\pi)^{-s}\Gamma(s)}\int_0^\infty x^{s-1}(\theta(ix)^2-1)dx$ where $\theta(z)^2 = \sum_{(a,b) \in \mathbb{Z}^2} e^{2i \pi z(a^2+b^2)})= (z/2i)^{-1} \theta(-4/z)^2$. $\endgroup$
    – reuns
    Commented Nov 18, 2017 at 21:58
  • $\begingroup$ In general to see if $F'(0) $ is well-defined where $F(s) =\sum_{n=1}^\infty a_n n^{-s}$ you need to find a convergent series $g(x)= \sum_{k=0}^\infty c_k x^{\alpha_k} (\log x)^{r_k}$ such that $\sum_{n=1}^\infty a_n e^{-nx}-\sum_{k=0}^\infty c_k x^{\alpha_k} (\log x)^{r_k}=o(1/\log^3 x)$ as $x \to 0$ ($-\alpha_k$ are the poles of $F(s)\Gamma(s)$) $\endgroup$
    – reuns
    Commented Nov 18, 2017 at 22:02
  • $\begingroup$ ncatlab.org/nlab/show/… with $\tau=i$ $\endgroup$
    – AHusain
    Commented Nov 18, 2017 at 23:53
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    $\begingroup$ This amounts again to the Kronecker limit formula. See, for example, this paper of Quine and Stong (A double Stirling formula): ams.org/journals/proc/1993-119-02/S0002-9939-1993-1164151-5 . $\endgroup$
    – Vesselin Dimitrov
    Commented Nov 19, 2017 at 2:16

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