Number Theoretic Correlation Inequalities for Dirichlet Densities Original Research Article

Our main discovery is the inequalityimagewhereA,Bare arbitrary sets of positive integers, (A, B)={(a, b) : aset membership, variantA, bset membership, variantB} is the set of largest common divisors, [A, B]={[a, ^;b] : aset membership, variantA, bset membership, variantB} is the set of least common multiples, andimagedenotes the lower Dirichlet density. It is much more general than our recent inequality for multiples of sets, which in turn is sharper than Behrendʹs well-known inequality. We also extend another recently discovered inequality, which does not seem to have number theoretic predecessors.