"Counting arithmetic objects: zeros of L-functions and integer partitions"

Abstract: Counting problems lie at the heart of number theory, be it the study of primes, class numbers or the number of integer partitions. One of the most difficult underlying questions here pertains to the distribution of the zeros of L-functions. This goes back to Riemann's seminal paper from 1859 where he studied the (Riemann) zeta function from an analytic view-point. We begin this talk with a brief overview of the distribution of zeros of these L-functions and their connections to various geometric and algebraic objects. In the later part, we discuss their influence on the asymptotic behavior of the partition function with parts concerning a Chebotarev condition, a natural generalization of partitions into prime in arithmetic progressions. We examine the error terms for these asymptotics in the context of the Riemann Hypothesis and improve Vaughan's result on partitions into primes. In connection to a result of Bateman and Erdos, we explore monotonicity of the partition function explicitly via an asymptotic formula.

Host: Matt Kerr