A cell complex in number theory
2011 (English)In: Advances in Applied Mathematics, ISSN 0196-8858, E-ISSN 1090-2074, Vol. 46, no 1-4, 71-85 p.Article in journal (Refereed) Published
Let Delta(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 beta(k)(Delta(n)) and of their sum. We show that Delta(n) has the homotopy type of a wedge of spheres, and that as n -> infinity S beta(k)(Delta(n)) = 2n/pi(2) + O(n(theta)), for all theta > 17/54, Furthermore, for fixed k, beta k(Delta(n)) similar to n/2logn (log log n)(k)/k!. As a number-theoretic byproduct we obtain inequalities partial derivative(k)(sigma(odd)(k+1)(n)) infinity S beta k((Delta) over tilde (n)) = n/3 + O(n(theta)), for all theta > 22/27.
Place, publisher, year, edition, pages
2011. Vol. 46, no 1-4, 71-85 p.
Mertens function, Liouville function, Multicomplex, Cellular realization
IdentifiersURN: urn:nbn:se:kth:diva-33991DOI: 10.1016/j.aam.2010.09.007ISI: 000290190600006ScopusID: 2-s2.0-79953722811OAI: oai:DiVA.org:kth-33991DiVA: diva2:418417
QC 201105232011-05-232011-05-232011-05-23Bibliographically approved