There is a well-known Perron formula, which connects a mean value of certain arithmetic function with its Dirichlet series:
$$ \sum\_{n\le x} f(n) = {1\over 2\pi i} \int\_{c-i\infty}^{c+i\infty} F(s) x^s s^{-1} ds, $$
where $F(s)=\sum\_{n=1}^\infty f(n) n^{-s}$, $c>\sigma\_f$, $F(s)$ is absolutely convergent for $\Re s > \sigma\_f$.


Is there any two-dimensional analog? Something of form 
$$ \sum\_{m,n\le x} f(m,n) = {1\over 2\pi i} \int\_{c-i\infty}^{c+i\infty} \int\_{c-i\infty}^{c+i\infty} F(z,w) x^{z+w} (zw)^{-1} dz dw $$
for $F(z,w) = \sum\_{m,n=1}^\infty f(m,n) m^{-z} n^{-w}$.



Two-dimensional Perron formula
This is really more of a comment, but I don't have the ability to comment.
The error term in the truncated two-dimensional Perron formula has been worked out in pain-staking details in the following paper of Balazard, et. al


<http://iml.univ-mrs.fr/~balazard/pdfdjvu/19.pdf>


See Proposition 5 and 6.


Of course, you probably should work with a smoothing, in which case these issues disappear. Without a smoothing the convergence issues in the two-dimensional Perron integral could get rather annoying.


