  Limit laws for ergodic averages with a power singularity over circle
rotations were first proved by Sinai and Ulcigrai, as well as Dolgopyat and
Fayad. In this paper, we prove limit laws with an estimate for the rate of
convergence for the sum $\sum_{n=1}^N f(n \alpha)/n^p$ in terms of a
$1$-periodic function $f$ with a power singularity of order $p \ge 1$ at
integers. Our results apply in particular to cotangent sums related to Dedekind
sums, and to sums of reciprocals of fractional parts, which appear in
multiplicative Diophantine approximation. The main tools are Schmidt's method
in metric Diophantine approximation, the Gauss-Kuzmin problem and the theory of
$\psi$-mixing random variables.
