A Converse Theorem for Double Dirichlet Series and Shintani Zeta Functions

A Converse Theorem for Double Dirichlet Series and Shintani Zeta Functions Nikolaos Diamantis and Dorian Goldfeld

A Converse Theorem for Double Dirichlet Series and Shintani Zeta Functions Nikolaos Diamantis and Dorian Goldfeld

Bhargava Integer Cubes and Weyl Group Multiple Dirichlet Series

We investigate the Shintani zeta functions associated to the prehomogeneous spaces, the example under consideration is the set of 2 × 2 × 2 integer cubes. We show that there are three relative invariants under a certain parabolic group action, they all have arithmetic nature and completely determine the equivalence classes. We show that the associated Shintani zeta function coincides with the A...

A CONVERSE THEOREM FOR DOUBLE DIRICHLET SERIES By NIKOLAOS DIAMANTIS and DORIAN GOLDFELD

We prove that a certain vector valued double Dirichlet series satisfying appropriate functional equations is a Mellin transform of a vector valued metaplectic Eisenstein series. We establish an analogous result for scalar double Dirichlet series of the type studied by Siegel.

Math 254a: Zeta Functions and L-series

Theorem 1.1. Let G be a group of Dirichlet characters modulo n, and KG the associated field. Fix a prime number p, and let H1 ⊂ H2 ⊂ G be the groups of χ such that χ(p) = 1 or χ(p) 6= 0 respectively. Then G/H2 ∼= IKG,p ⊂ Gal(KG/Q), and G/H1 ∼= DKG,p (note that these are non-canonical isomorphisms). We are now ready to prove the theorem on factoring Dedekind zeta functions of sub-cyclotomic fiel...