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.