Analytic continuation of $ f(x,y)=\sum_{n=1}^{\infty}\sum_{k=1}^{\infty} e^{-(k^xn^y)}$

Question

I was thinking about adding up points on a particular surface and I derived the function $f(x,y).$

How do you maximally analytically continue $f(x,y)$?

$$ f(x,y)=\sum_{n=1}^{\infty}\sum_{k=1}^{\infty} e^{-(k^xn^y)}$$

I think $f(x,y)$ converges for $\Re(x,y)>1$ but I'm not sure. This is partly motivated by the question Analytic continuation.

I'm thinking about using a Cahen-Mellin integral with double integrals and then proceeding using the techniques in the linked question.

I'm interested in any literature where this function appears.