Density Inequalities for Sets of Multiples Original Research Article

For finite sets A, B subset of image, the set of positive integers, consider the set of least common multiples [A, B] = {[a, b]: a set membership, variant A, b set membership, variant B}, the set of largest common divisors (A, B) = {(a, b): a set membership, variant A, b set membership, variant B}, the set of products A × B = {a · b: a set membership, variant A, b set membership, variant B}, and the sets of their multiples M(A) = A × image, M(B), M[A, B], M(A, B), and M(A × B), resp. Our discoveries are the inequalities dM(A, B) dM[A, B] ≥ dM(A) · dM(B) ≥ dM(A × 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.