Let K be a degree n extension of Q. How many orders are contained the ring of integers of K with index at most B?
This question about counting orders fits nicely into the theory of zeta functions of groups and rings. Given an infinite group G we can define a zeta function that is a generating function for the number of subgroups of index k. When G = Z^n this is a product of Riemann zeta functions. The analogous zeta function for subrings of a given ring is generally much more complicated, even in simple examples.
These zeta functions have Euler product expansions with local factors that can be understood as p-adic integrals. Even in cases where these integrals have not been computed exactly analytic properties of zeta functions give information about these counting problems. We will discuss joint work with Ramin Takloo-Bighash where we use this approach to study orders in quintic number fields.