  Let $U\in U(N)$ be a random unitary matrix of size $N$, distributed with
respect to the Haar measure on $U(N)$. Let $P(z)=P_U(z)$ be the characteristic
polynomial of $U$. We prove that for $z$ close to the unit circle, $
\frac{P'}{P}(z) $ can be approximated using zeros of $P$ very close to $z$,
with a typically controllable error term. This is an analogue of a result of
Selberg for the Riemann zeta-function. We also prove a mesoscopic central limit
theorem for $ \frac{P'}{P}(z) $ away from the unit circle, and this is an
analogue of a result of Lester for zeta.
