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/