Date of Degree

5-2018

Document Type

Dissertation

Degree Name

Ph.D.

Program

Mathematics

Advisor

Gautam Chinta

Committee Members

Melvyn Nathanson

Cormac O'Sullivan

Subject Categories

Number Theory

Keywords

Fourier series, Hecke L-function, Totally positive integer, Trace

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 number field is quadratic, Beck also proved a mean value result using the continued fraction expansions of quadratic irrationals. We generalize Beck’s result to higher moments. 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 Grossencharacter L -functions.

Included in

Number Theory Commons

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