Abstract: In this talk we will discuss Montgomery's pair correlation conjecture for the zeros of the Riemann zeta function. This is a deep spectral conjecture, closely related to several arithmetic conjectures on the distribution of primes. For example, even assuming a strong form of the twin prime conjecture, one would only resolve Montgomery's conjecture in a limited range. Yet, building on work of Goldston and Gonek from the late 1990s, we will present a recent conditional lower bound on the Fourier transform of Montgomery's pair correlation function, valid under milder hypotheses. The new technical ingredient is a correlation estimate for the Selberg sieve weights, for which the level of support of the weights lies beyond the classical square-root barrier.