Gautam Chinta: Zeta functions of rings

We will describe the constructions of three types of Dirichlet series associated with commutative rings. Analytic properties of these series can be used to solve various counting problems. * The first count ideals in a fixed ring R, ordered by norm. When R is the maximal order of a number field, this zeta function coincides with the Dedekind zeta function of the number field. * The second --- introduced by Shintani --- counts rings of a fixed rank n (n=2,3) ordered by discriminant. * The third, which is a Dirichlet series in two complex variable, interpolates between the the first two and displays a rich structure.