Tianyi Mao
Published 2016-06-13Version 1
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.
On the distribution of Atkin and Elkies primes for reductions of elliptic curves on average
On the distribution of matrix elements for the quantum cat map
Gaps between zeros of $ζ(s)$ and the distribution of zeros of $ζ'(s)$
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