  We prove some results on the behavior of infinite sums of the form $\Sigma
f\circ T^n(x)\frac{1}{n}$, where $T:S^1\to S^1$ is an irrational circle
rotation and $f$ is a mean-zero function on $S^1$. In particular, we show that
for a certain class of functions $f$, there are Liouville $\alpha$ for which
this sum diverges everywhere. We also show that there are Liouville $\alpha$
for which the sum converges everywhere.
