Kelly Isham : Zeta functions and asymptotics related to subrings in Z^n

Posted by Jacob Mayle , part of the Number Theory Seminar.

At

Feb. 25, 2022, 1 p.m.

In

Zoom

Abstract

We can define a zeta function of a group (or ring) to be the Dirichlet series associated to the sequence that counts the number of subgroups (or subrings) of a given index. The subgroup zeta function over Z^n is well-understood, as is the asymptotic growth of subgroups in Z^n. Much less is known about the subring zeta function over Z^n and the asymptotic growth of subrings in Z^n. In this talk, we discuss the progress toward answering this question and we give new lower bounds on the asymptotic growth of subrings in Z^n. We then define a similar zeta function corresponding to subrings of corank at most k in Z^n. While the proportion of subgroups in Z^n of corank k is positive for each k, we show this is not the case for subrings in Z^n of corank k when n is sufficiently larger than k. If there is time, we will make connections to orders in number fields. Part of this work is joint with Nathan Kaplan.

Zoom link: https://uic.zoom.us/j/88173268700?pwd=aEhmTGpSOVhidWE4L1VWUnNhNVlvUT09