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Am interested in how one could regularize the following sum $\sum_{m,n = 1, \infty} \sqrt{m^2 + n^2}$. Would preferably want this in the $\epsilon$ expansion regularization as talked below where a series $\sum_{n=1,\infty} n$ is regularized by introducing $\sum_{n=1,\infty} n e^{-\epsilon n}$

http://blog.wolfram.com/2014/08/06/the-abcd-of-divergent-series/

vaibhav wasnik
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  • Don't know if this helps, but the number $a(n)$ of lattice points with norm $n$ is http://oeis.org/A046109. The answer would be something like $\frac{1}{4} \sum_n n a(n)$. – user76284 Jul 19 '19 at 23:49
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    You could also try analytically continuing $\sum_{m,n=1}^\infty (m^2+n^2)^s$ where it converges to $s=1/2$. – user76284 Jul 20 '19 at 00:25

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