Correlations of sieve weights and distributions of zeros

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.

number theory

Audience: researchers in the topic

London number theory seminar

Series comments: For reminders, join the (very low traffic) mailing list at mailman.ic.ac.uk/mailman/listinfo/london-number-theory-seminar

For a record of talks predating this website see: wwwf.imperial.ac.uk/~buzzard/LNTS/numbtheo_past.html