Simplicial complexes associated to certain subsets of natural numbers   and its applications to multiplicative functions
  We call a set of positive integers closed under taking unitary divisors a unitary ideal. It can be regarded as a simplicial complex. Moreover, a multiplicative arithmetical function on such a set corresponds to a function on the simplicial complex with the property that the value on a face is the product of the values at the vertices of that face. We use this observation to solve the following problems:   1) Let r be a positive integer and c a real number. What is the maximum value that \sum_{s \in S}g(s) can obtain when S is a unitary ideal containing precisely r prime powers, and g is the multiplicative function determined by g(s)=c when s \in S is a prime power?   2) Suppose that g is a multiplicative function which is \ge 1, and that we want to find the maximum of g(i) when 1 \le i \le n. At how many integers do we need to evaluate g? 
Jan Snellman