I am trying to manipulate this summation such that I have a summation dependent only on $i,j,k$ and any other outside influence or dependence on $B$ is left outside of this sum:
$$\sum_{i=1}^{\infty}\sum_{j=1}^{\infty}\sum_{k=1}^{\infty}e^{-B\sqrt{i^2+j^2+k^2}}$$
This could be rewritten: $$\sum_{i=1}^{\infty}\sum_{j=1}^{\infty}\sum_{k=1}^{\infty}\left(e^{-B}\right)^{\sqrt{i^2+j^2+k^2}}$$ Or $$\sum_{i=1}^{\infty}\sum_{j=1}^{\infty}\sum_{k=1}^{\infty}\left(e^{\sqrt{i^2+j^2+k^2}}\right)^{-B}$$ Or multiple other ways, but you get the idea.
The reason I want to do this is because I could potentially handle the summation on $(i,j,k)$ independently, which would be great!
I've looked at trying things like manipulating with $Ln$ functions but that always seems to diverge...
Also, it is not possible to use this replacement: $$r\equiv\sqrt{i^2+j^2+k^2}$$ and then $$\sum_{i=1}^{\infty}e^{-B\cdot{r}}$$ Because $r$ is not guaranteed to be an integer (it usually isn't).
Any tricks/thoughts are much appreciated! (Also, $B$ is positive and real)