Is there a closed form for:$$ f(2,2)=\sum_{n=1}^\infty \sum_{k=1}^\infty \exp\big(- n^2k^2 \big)? $$
Note that I'm defining a function: $$f(x,y)=\sum_{n=1}^\infty \sum_{k=1}^\infty\exp\big(-n^xk^y\big)$$ and seeking a closed form for:
$f(2,2)\approx .404$
In the case $f(1,1)$ there is indeed a closed form in terms of something called a q-Polygamma function which can sometimes be related to Jacobi theta functions.
This is interesting because $f(2)$ can be written in terms of a Jacobi third theta function.