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arXiv · 1810.00382

The Hlawka Zeta Function as a Respectable Object

Michael Montoro
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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.

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Michael Montoro. 2018-09-30. The Hlawka Zeta Function as a Respectable Object. https://arxiv.org/abs/1810.00382

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