The Hlawka Zeta Function as a Respectable Object

TL;DR

The paper explores the Hlawka Zeta Function, a geometric Dirichlet series related to lattice point counting, analyzing its properties through Eisenstein Series and studying specific shapes like circles and squares.

Contribution

It presents a new integral representation of the Hlawka Zeta Function using Eisenstein Series and investigates its properties for various geometric regions.

Findings

Derived the Hlawka Zeta Function as a sum of Eisenstein Series.

Analyzed functional equations for specific shapes such as circles and squares.

Posed conjectures on the general properties of the Hlawka Zeta Function.

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.