Princeton University Mathematics Department Colloquium

We review the theory of multiple Dirichlet series which are Dirichlet series in several complex variables having analytic continuation with finitely many polar divisors and satisfying a finite group of functional equations. Converse theorems state that if a Dirichlet series and all its twists satisfy functional equations of the right type then the Dirichlet series is modular, i.e., it is the Mellin transform of an automorphic form. We shall present a new converse theorem for Dirichlet series in two complex variables. As an application we will show that the Shintani zeta function associated to a certain pre homogeneous vector space is in fact a Weyl group multiple Dirichlet series of type \(A_2\). This is joint work with Nikos Diamantis.