Well-rounded zeta-function of planar arithmetic lattices

TL;DR

This paper studies the zeta-function of well-rounded sublattices in planar arithmetic lattices, revealing its convergence properties and providing bounds on the count of such sublattices with bounded index.

Contribution

It establishes the abscissa of convergence and pole order of the zeta-function, improving previous results, and derives an asymptotic bound for the number of well-rounded sublattices.

Findings

Zeta-function has abscissa of convergence at s=1 with a pole of order 2.

Number of well-rounded sublattices of index ≤ N is O(N log N).

Provides a description of integral well-rounded sublattices and analyzes their zeta-functions.

Abstract

We investigate the properties of the zeta-function of well-rounded sublattices of a fixed arithmetic lattice in the plane. In particular, we show that this function has abscissa of convergence at $s=1$ with a real pole of order 2, improving upon a recent result of S. Kuehnlein. We use this result to show that the number of well-rounded sublattices of a planar arithmetic lattice of index less or equal $N$ is $O(N \log N)$ as $N \to \infty$. To obtain these results, we produce a description of integral well-rounded sublattices of a fixed planar integral well-rounded lattice and investigate convergence properties of a zeta-function of similarity classes of such lattices, building on some previous results of the author.