2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/163373We consider Birkhoff sums of functions with a singularity of type 1/x over rotations and prove the following limit theorem. Let $S_N= S_N(α,x)$ be the N^th non-renormalized Birkhoff sum, where $x in [0,1)$ is the initial point, $α\in [0,1)$ is the rotation number and $(α, x)$ are uniformly distributed. We prove that $S_N/N$ has a joint limiting distribution in $(α,x)$ as N tends to infinity. As a corollary, we get the existence of a limiting distribution for certain trigonometric sums.24 pages, some typos corrected, final version to appear in ``Probabilistic and Geometric Structures in Dynamics'', edited by K. Burns, D. Dolgopyat, and Ya. Pesis, American Mathematical Society, Contemporary Mathematics SeriesDynamical Systemstext