Sums over primitive sets with a fixed number of prime factors

Abstract:

A primitive set is one in which no element of the set divides another. Erdős conjectured that the sum \begin{equation*} f(A) := \sum _{n \in A} \frac {1}{n \log n} \end{equation*} taken over any primitive set $A$ would be greatest when $A$ is the set of primes. More recently, Banks and Martin have generalized this conjecture to claim that, if we let $\mathbb {N}_k$ represent the set of integers with precisely $k$ prime factors (counted with multiplicity), then we have $f(\mathbb {N}_1) > f(\mathbb {N}_2) > f(\mathbb {N}_3) > \cdots$. The first of these inequalities was established by Zhang; we establish the second. Our methods involve explicit bounds on the density of integers with precisely $k$ prime factors. In particular, we establish an explicit version of the Hardy-Ramanujan theorem on the density of integers with $k$ prime factors.