Number Theory Web Seminar

Jeffrey C. Lagarias, The Alternative Hypothesis and Point Processes
(University of Michigan)

Thursday, October 6, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, October 7, 2022 (2am AEDT, 4am NZST)

Abstract: The Alternative Hypothesis concerns a hypothetical and unlikely picture of how zeros of the Riemann zeta function are spaced. It asks that nearly all normalized zero spacings be near half-integers. This possible zero distribution is incompatible with the GUE distribution of zero spacings. Ruling it out arose as an obstacle to the long-standing problem of proving there are no exceptional zeros of Dirichlet L-functions. The talk describes joint work with Brad Rodgers, that constructs a point process realizing Alternative Hypothesis type statistics, which is consistent with the known results on correlation functions for spacings of zeta zeros. (A similar result was independently obtained by Tao with slightly different methods.) The talk reviews point process models and presents further results on the general problem of to what extent two point processes, a continuous one on the real line, the other a discrete one on a lattice aZ, can mimic each other in the sense of having perfect agreement of all their correlation functions when convolved with bandlimited test functions of a given bandwidth B.