Date of Degree
Fourier series, Hecke L-function, Totally positive integer, Trace
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.
Mao, Tianyi, "The Distribution of Totally Positive Integers in Totally Real Number Fields" (2018). CUNY Academic Works.