  We consider Birkhoff sums of functions with a singularity of type 1/x over
rotations and prove the following limit theorem. Let $S_N= S_N(\alpha,x)$ be
the N^th non-renormalized Birkhoff sum, where $x in [0,1)$ is the initial
point, $\alpha\in [0,1)$ is the rotation number and $(\alpha, x)$ are uniformly
distributed. We prove that $S_N/N$ has a joint limiting distribution in
$(\alpha,x)$ as N tends to infinity. As a corollary, we get the existence of a
limiting distribution for certain trigonometric sums.
