math.NT

# Title: 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.
 Comments: 27 Pages, Submitted as Undergraduate Honors Thesis to State University of New York at Buffalo, under advisement of Joseph Hundley Subjects: Number Theory (math.NT) MSC classes: 11M41 Cite as: arXiv:1810.00382 [math.NT] (or arXiv:1810.00382v2 [math.NT] for this version)

## Submission history

From: Michael Montoro [view email]
[v1] Sun, 30 Sep 2018 13:57:20 GMT (23kb)
[v2] Tue, 30 Jun 2020 13:28:58 GMT (23kb)

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