Abstract
Our main discovery is the inequalityD(A, BD[A, B]⩾DADB,whereA,Bare arbitrary sets of positive integers, (A, B)={(a, b):a∈A, b∈B} is the set of largest common divisors, [A, B]={[a, ^;b]:a∈A, b∈B} is the set of least common multiples, andDdenotes the lower Dirichlet density. It is much more general than our recent inequality for multiples of sets, which in turn is sharper than Behrend's well-known inequality. We also extend another recently discovered inequality, which does not seem to have number theoretic predecessors.
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