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 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.
© 2013 American Mathematical Society
Fukshansky, Lenny. "Well-Rounded Zeta-Function of Planar Arithmetic Lattices." Proceedings of the American Mathematical Society 142.2 (2014): 369-380. doi: 10.1090/S0002-9939-2013-11820-4