WELL-ROUNDED ZETA-FUNCTION OF PLANAR ARITHMETIC LATTICES

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 Kuhnlein. 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 -> infinity. 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.