Number Theory Seminar: Abhishek Jha (UIUC)

Abstract: Fifty years ago, Gallagher showed that prime counts in intervals of length λ log(x) are asymptotically Poisson for fixed λ, assuming the Hardy-Littlewood conjectures. In this talk, I will discuss what happens when λ is allowed to grow with x. Under suitable uniform Hardy-Littlewood hypotheses, we obtain a growing range in which the Poisson prediction remains valid and identify a phase transition beyond which it breaks down. The talk will focus on the probabilistic and sieve-theoretic ideas underlying these results.