کد مقاله کد نشریه سال انتشار مقاله انگلیسی نسخه تمام متن
4595169 1335801 2008 35 صفحه PDF دانلود رایگان
عنوان انگلیسی مقاله ISI
On distribution of well-rounded sublattices of Z2
موضوعات مرتبط
مهندسی و علوم پایه ریاضیات اعداد جبر و تئوری
پیش نمایش صفحه اول مقاله
On distribution of well-rounded sublattices of Z2
چکیده انگلیسی

A lattice is called well-rounded if its minimal vectors span the corresponding Euclidean space. In this paper we completely describe well-rounded full-rank sublattices of Z2, as well as their determinant and minima sets. We show that the determinant set has positive density, deriving an explicit lower bound for it, while the minima set has density 0. We also produce formulas for the number of such lattices with a fixed determinant and with a fixed minimum. These formulas are related to the number of divisors of an integer in short intervals and to the number of its representations as a sum of two squares. We investigate the growth of the number of such lattices with a fixed determinant as the determinant grows, exhibiting some determinant sequences on which it is particularly large. To this end, we also study the behavior of the associated zeta function, comparing it to the Dedekind zeta function of Gaussian integers and to the Solomon zeta function of Z2. Our results extend automatically to well-rounded sublattices of any lattice AZ2, where A is an element of the real orthogonal group O2(R).

ناشر
Database: Elsevier - ScienceDirect (ساینس دایرکت)
Journal: Journal of Number Theory - Volume 128, Issue 8, August 2008, Pages 2359-2393