Well-rounded zeta-function of planar arithmetic lattices

Fukshansky, Lenny

doi:10.1090/S0002-9939-2013-11820-4

Well-rounded zeta-function of planar arithmetic lattices
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Proc. Amer. Math. Soc. 142 (2014), 369-380 Request permission

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 result of Stefan Kühnlein. We use this result to show that the number of well-rounded sublattices of a planar arithmetic lattice of index less than or equal to $N$ 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 the results of a paper by Glenn Henshaw, Philip Liao, Matthew Prince, Xun Sun, Samuel Whitehead, and the author.

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