  We investigate the distribution of the Riemann zeta-function on the line
$\Re(s)=\sigma$. For $\tfrac 12 < \sigma \le 1$ we obtain an upper bound on the
discrepancy between the distribution of $\zeta(s)$ and that of its random
model, improving results of Harman and Matsumoto. Additionally, we examine the
distribution of the extreme values of $\zeta(s)$ inside of the critical strip,
strengthening a previous result of the first author.
  As an application of these results we obtain the first effective error term
for the number of solutions to $\zeta(s) = a$ in a strip $\tfrac12 < \sigma_1 <
\sigma_2 < 1$. Previously in the strip $\tfrac 12 < \sigma < 1$ only an
asymptotic estimate was available due to a result of Borchsenius and Jessen
from 1948 and effective estimates were known only slightly to the left of the
half-line, under the Riemann hypothesis (due to Selberg) and to the right of
the abscissa of absolute convergence (due to Matsumoto). In general our results
are an improvement of the classical Bohr-Jessen framework and are also
applicable to counting the zeros of the Epstein zeta-function.
