The Distribution of Integers in a Totally Real Cubic Field

TL;DR

This paper investigates how integers are distributed in totally real cubic fields, linking the asymptotic behavior of certain sums to the structure of the unit group and expressing main terms via Gr"ossencharacter L-functions.

Contribution

It generalizes previous methods to cubic fields and relates distribution properties to the unit group structure and L-functions.

Findings

Asymptotic behavior of weighted sums is characterized

Main term expressed through Gr"ossencharacter L-functions

Distribution of integers linked to unit group structure

Abstract

Hecke studies the distribution of fractional parts of quadratic irrationals with Fourier expansion of Dirichlet series. This method is generalized by Behnke and Ash-Friedberg, to study the distribution of the number of totally positive integers of given trace in a general totally real number field of any degree. When the field is cubic, we show that the asymptotic behavior of a weighted Diophantine sum is related to the structure of the unit group. The main term can be expressed in terms of Gr\"ossencharacter $L$-functions.