A cell complex in number theory

Abstract

Let Δn 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(Δn) and of their sum. We show that Δn has the homotopy type of a wedge of spheres, and that as n→∞∑βk(Δn)=2nπ2+O(nθ),for all θ>1754. Furthermore, for fixed k,βk(Δn)∼n2logn(loglogn)kk!. As a number-theoretic byproduct we obtain inequalities∂k(σk+1odd(n))⩽σkodd(n/2), where σkodd(n) denotes the number of odd squarefree integers⩽n with k prime factors, and ∂k is a certain combinatorial shadow function.We also study a CW complex Δ˜n that extends the previous simplicial complex. In Δ˜nall numbers ⩽n correspond to cells and its Euler characteristic is the summatory Liouville function. This cell complex Δ˜n is shown to be homotopy equivalent to a wedge of spheres, and as n→∞∑βk(Δ˜n)=n3+O(nθ),for all θ>2227.