A cell complex in number theory

Abstract

Let De_n 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 and of their sum. We show that De_n has the homotopy type of a wedge of spheres, and that as n tends to infinity: $$\sum \be_k(\De_n) = \frac{2n}{\pi^2} + O(n^{\theta}),\;\; \mbox{for all} \theta > \frac{17}{54}.$$ We also study a CW complex tDe_n that extends the previous simplicial complex. In tDe_n all numbers up to n correspond to cells and its Euler characteristic is the summatory Liouville function. This cell complex is shown to be homotopy equivalent to a wedge of spheres, and as n tends to infinity: $$\sum \be_k(\tDe_n) = \frac{n}{3} + O(n^{\theta}),\;\; \mbox{for all} \theta > \frac{22}{27}.$$