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 Szegő, 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

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... atrix. 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.