PUB - Publikationen an der Universität Bielefeld

Our main discovery is the inequality (D) under bar(A,B) (D) under bar[A, B] greater than or equal to (D) under bar A (D) under bar B, where A, B are arbitrary sets of positive integers. (A, B) = {(a, b):a epsilon A, b epsilon B} is the set of largest common divisors, [A, B] = {[a, b]: a is an element of A, b is an element of B} is the set of least common multiples. and (D) under bar denotes 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. (C) 1997 Academic Press.