  We unconditionally prove a central limit theorem for linear statistics of the
zeros of the Riemann zeta function with diverging variance. Previously,
theorems of this sort have been proved under the assumption of the Riemann
hypothesis. The result mirrors central limit theorems in random matrix theory
that have been proved by Szeg\H{o}, Spohn, and Soshnikov among others, and
therefore provides support for the view that the zeros of the zeta function are
distributed like the eigenvalues of a random matrix.
  A key ingredient in our proof is a simple bootstrapping of classical zero
density estimates of Selberg and Jutila for the zeta function, which may be of
independent interest.
