The Hlawka Zeta Function as a Respectable Object

Abstract

The Hlawka Zeta Function is a Dirichlet series defined geometrically which provides an integral representation of the number of lattice points contained in the dilation $tD$ for some star shaped region $D\subset {\mathbb{R}}^2$ and some real number $t\in {\mathbb{R}}^{+}$. We give an overview of this construction and integral representation before giving the Hlawka Zeta function as a sum of Eisenstein Series acting on $K$-finite vectors multiplied by Fourier coefficients depending on $D$. We then study the case of $D$ as an circle, ellipse, and then square to study functional equations and "fibers" of this object, and pose conjectures regarding these properties in general.

27 Pages, Submitted as Undergraduate Honors Thesis to State University of New York at Buffalo, under advisement of Joseph Hundley