# Zeta Functions and Number Fields

My main research has been a long exploration into how the properties of a zeta function defined by Mikio Sato and greatly developed by Takuro Shintani [#!SatoM1970!#,#!SatoShintani1974!#] may be used to study the discriminants and class numbers of algebraic number fields of low degree. These zeta functions are associated to certain representations of algebraic groups which were called prehomogeneous by Sato. My thesis work and first several papers concerned the prehomogeneous space of binary cubic forms which was first thoroughly explored by Shintani [#!Shintani1972!#] over the base field . I used Weil's framework for zeta functions and Eisenstein series presented in [#!Weil1964!#,#!Weil1965!#] to generalize Shintani's work to the case where the base field is an arbitrary number field and then streamlined the proof based on the zeta function of the density of discriminants of cubic extensions of an arbitrary number field and the mean-value of the two-class-number of quadratic extensions of a number field. My friend and younger graduate student Boris Datskovsky joined me on this work towards the end in the papers [#!Wright1982!#,#!Wright1985!#,#!DatskovskyWright1986!#,#!DatskovskyWright1988!#,#!Wright1987!#].

There are several aspects of this work that remain active today. The original theorem of Davenport and Heilbronn which Datskovsky and I reproved and generalized says that the number of cubic fields with discriminant not greater than in absolute value behaves like

This comes out of the zeta function theory in much the same way that the Prime Number Theorem emerged from the Riemann zeta function. However, Shintani's zeta function theory offered much more than this. The zeta function is a function of a complex variable which has a simple pole at and this ultimately leads to the Davenport-Heilbronn theorem. However, there is a second pole at which suggests a precise error term