  We prove a multidimensional extension of Selberg's central limit theorem for
$\log\zeta$, in which non-trivial correlations appear. In particular, this
answers a question by Coram and Diaconis about the mesoscopic fluctuations of
the zeros of the Riemann zeta function.
  Similar results are given in the context of random matrices from the unitary
group. This shows the correspondence $n \leftrightarrow \log t$ not only
between the dimension of the matrix and the height on the critical line, but
also, in a local scale, for small deviations from the critical axis or the unit
circle.
