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Well-rounded zeta-function of planar arithmetic lattices


Author: Lenny Fukshansky
Journal: Proc. Amer. Math. Soc. 142 (2014), 369-380
MSC (2010): Primary 11H06, 11H55, 11M41, 11E45
DOI: https://doi.org/10.1090/S0002-9939-2013-11820-4
Published electronically: October 17, 2013
MathSciNet review: 3133979
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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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Additional Information

Lenny Fukshansky
Affiliation: Department of Mathematics, Claremont McKenna College, 850 Columbia Avenue, Claremont, California 91711
Email: lenny@cmc.edu

DOI: https://doi.org/10.1090/S0002-9939-2013-11820-4
Keywords: Arithmetic lattices, integral lattices, well-rounded lattices, Dirichlet series, zeta-functions
Received by editor(s): January 12, 2012
Received by editor(s) in revised form: March 18, 2012
Published electronically: October 17, 2013
Additional Notes: The author was partially supported by a grant from the Simons Foundation (#208969) and by the NSA Young Investigator Grant #1210223.
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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