کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
6419689 | 1631647 | 2011 | 15 صفحه PDF | دانلود رایگان |

Let În be the simplicial complex of squarefree positive integers less than or equal to n ordered by divisibility. It is known that the asymptotic rate of growth of its Euler characteristic (the Mertens function) is closely related to deep properties of the prime number system.In this paper we study the asymptotic behavior of the individual Betti numbers βk(În) and of their sum. We show that În has the homotopy type of a wedge of spheres, and that as nâââβk(În)=2nÏ2+O(nθ),for all θ>1754. Furthermore, for fixed k,βk(În)â¼n2logn(loglogn)kk!. As a number-theoretic byproduct we obtain inequalitiesâk(Ïk+1odd(n))⩽Ïkodd(n/2), where Ïkodd(n) denotes the number of odd squarefree integers ⩽n with k prime factors, and âk is a certain combinatorial shadow function.We also study a CW complex ÎËn that extends the previous simplicial complex. In ÎËnall numbers ⩽n correspond to cells and its Euler characteristic is the summatory Liouville function. This cell complex ÎËn is shown to be homotopy equivalent to a wedge of spheres, and as nâââβk(ÎËn)=n3+O(nθ),for all θ>2227.
Journal: Advances in Applied Mathematics - Volume 46, Issues 1â4, January 2011, Pages 71-85