2010/09/06 by Greg Martin, Carl Pomerance, Martin, Greg +1
Mathematics · #FOS: Mathematics #Number Theory (math.NT) #math.NT
paper · pdf · doi:10.48550/arxiv.1009.1014
7 pages. Revision includes a strengthening of Theorem 1: an upper bound for S(x) of the same order of magnitude as the lower bound is now established
arxiv created 2010/10/27 · arxiv updated 2010/10/28
A set of positive integers is said to be primitive if no element of the set is a multiple of another. If S is a primitive set and S(x) is the number of elements of S not exceeding x, then a result of Erd\H os implies that ∫2^∞ (S(t)/t2log t) dt converges. We establish an approximate converse to this theorem, showing that if F satisfies some mild conditions and ∫2^∞ (F(t)/t2log t) dt converges, then there exists a primitive set S with S(x) ≫ F(x).