Density inequalities for sets of multiples
For finite sets A, B subset of N the set of positive integers, consider the set of least common multiples [A, B] = {[a, b]: a is an element of A, b is an element of B}, the set of largest common divisors (A, B) = {(n, b): a is an element of A, b is an element of B}, the set of products A x B = {a . b: a is an element of A, b is an element of B}, and the sets of their multiples M (A) = A x N, M(B), M[A, B], M(A, B), and M(A x B), resp. Our discoveries are the inequalities dM(A, B)dM[A, B]greater than or equal to dM(A). dM(B)greater than or equal to dM(A x B), where d denotes the asymptotic density. The first inequality is by the factor dM(A,B) sharper than Behrend's well-known inequality. This in turn is a generalisation of an earlier inequality of Rohrbach and Heilbronn, which settled a conjecture of Hasse concerning an identity due to Direchlet. Our second inequality does not seem to have predecessors. (C) 1995 Academic Press, Inc.
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