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Is it possible to express the following sum in closed form? $$S(p)=\sum_{n=-\infty}^{\infty}\sum_{m=-\infty}^{\infty}{\left(n^2+m^2\right)^{\large-{p}}}$$ where the point $(n,m)=(0,0)$ is not taken into account in the sum (i.e. $(n,m)\in \mathbb{Z}^2/\{(0,0)\}$).

In the book Lattice Sums Then and Now by Borwein et al. it is stated (page 66, Appendix 1.7), that $S(p)=4\beta(p)\zeta(p)$, with $\beta$ and $\zeta$ the Dirichlet beta and Riemann zeta functions.

However I cannot find any derivation of this result. Any thoughts?

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If $\chi_4$ denotes the unique non-trivial Dirichlet character modulo 4, then it is proved that for $k \geq 1$,

$$ r_2(k) := \#\{ (m,n)\in\mathbb{Z}^2 : m^2+n^2=k\} = 4 \sum_{d \mid k} \chi_4(d). $$

(I do not remember the proof of this fact, but the keyword sum of squares function may be useful for finding out a reference.)

Now utilizing this, we have

\begin{align*} S(p) = \sum_{k=1}^{\infty} k^{-p} r_2(k) &= 4 \sum_{k=1}^{\infty} \left( \sum_{d\mid k} \chi_4(d) \right) k^{-p} \\ &= 4 \left( \sum_{d=1}^{\infty} \chi_4(d) d^{-p} \right) \left( \sum_{l=1}^{\infty} l^{-p} \right) = 4 \beta(p)\zeta(p). \end{align*}

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  • $\begingroup$ I believe there is a misprint in the first expression: the sum should be over $d\mid k$. $\endgroup$ – user Feb 19 '18 at 14:27
  • $\begingroup$ @user355705, Thank you, I fixed the typo. $\endgroup$ – Sangchul Lee Feb 19 '18 at 23:10
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    $\begingroup$ Nice and simple proof +1. The proof for sum of squares function can be provided by using Jacobi elliptic/theta functions and their Fourier series, and there is also a pure number theoretic proof but don't remember any reference for it. $\endgroup$ – Paramanand Singh Feb 21 '18 at 14:23
  • $\begingroup$ @ParamanandSingh, Thank you for the informative comment. As for the number-theoretic proof, now I faintly remember that this is related to factorization in $\mathbb{Z}[i]$. $\endgroup$ – Sangchul Lee Feb 22 '18 at 2:02

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