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The expression is as follows:

$\sum_{x=0}^{\infty}\sum_{y=0}^{\infty} \exp(-\sqrt{x^2+y^2})$

I have thought about using Taylor approximation to get started but it doesn't seem to get me anywhere.

Any hints on how to get started will be appreciated. Thank you in advance.

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I assume that $x$ and $y$ are integral, in which case your sum is one quarter of the sum

$$\sum_{r=1}^\infty \exp(-\sqrt{r}) s(r),$$ where $s(r)$ is the number of representations of $r$ as the sum of two squares. Clearly, $\sum_{r=1}^R s(r) \sim \pi r,$ which means you can estimate the original sum via summation by parts, followed by approximating the sum by the integral.

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  • $\begingroup$ In your notation the area of a circle is $\pi r$. The sum you get with the aporoximation can be computed exactly by summation by parts again. $\endgroup$
    – Will Sawin
    Commented May 16, 2015 at 23:56
  • $\begingroup$ @WillSawin agreed on both points. I fixed the first, not the second, since it still doesn't actually give any sort of closed form for the sum. $\endgroup$
    – Igor Rivin
    Commented May 17, 2015 at 0:00
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    $\begingroup$ The sum converges fast enough to be computed efficiently to any desired accuracy as it stands. But I suspect that there might be no closed form. $\endgroup$
    – Noam D. Elkies
    Commented May 17, 2015 at 1:39
  • $\begingroup$ @NoamD.Elkies Yes, indeed Mathematica is happy to evaluate it, but it would be interesting if there were a closed form (of course, it depends on what you mean by "closed"). $\endgroup$
    – Igor Rivin
    Commented May 17, 2015 at 14:56

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