I am wondering if a closed form exists to the lattice sum $$S(a)= \sum_{m=-\infty}^{\infty} \sum_{n=-\infty}^{\infty} \frac 1 {(a^2+m^2+n^2)^{3/2}}$$ I am also interested in replacing $m^2+n^2$ with a general positive definite quadratic form.
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$\begingroup$ For the record, I know the large $a$ asymptotics (via Riemann Sums) and I've consulted Lattice Sums Then and Now by Borwein et al. $\endgroup$– AndrewBernoffAug 31, 2017 at 15:50
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1$\begingroup$ $=\frac{\pi^{3/2}}{\Gamma(3/2)} \int_0^\infty e^{-\pi a^2 x} \theta(x)^2 x^{1/2}dx, \theta(x) = \sum_{n} e^{- \pi n^2 x}$ $\endgroup$– reunsAug 31, 2017 at 23:57
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2$\begingroup$ When $a = 0$, this is (morally) the value of the zeta function of $\mathbb{Q}[i]$ at $s = 3/2$. Since it's non-critical, it's probably just a mysterious number. $\endgroup$– WhatsUpSep 1, 2017 at 2:49
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$\begingroup$ I worry (may need to accept) that variants of @reuns expression may be as good as it gets. I get $\theta(x) \sim 1/\sqrt{x}$ near $x=0$, so the integral converges. $\endgroup$– AndrewBernoffSep 1, 2017 at 15:36
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