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In this paper, we study how dense a multiplicative basis of order h for Z + can be, improving on earlier results. Upon introducing the notion of a multiplicative complement, we present some tight density bounds. IntroductionLet Z + denote the set of positive integers. For A ⊆ Z + and h ∈ Z + , the multiplicative representation function S A,h (n) of order h counts the ordered representations of n ∈ Z + as a product of h elements of A; that is, we define S A,h (n) = |{(a 1 , . . . , a h ) : a i ∈ A and a 1 • • • a h = n}|.