SUMS OVER PRIMITIVE SETS WITH A FIXED NUMBER OF PRIME FACTORS.

A primitive set is one in which no element of the set divides another. Erdos conjectured that the sum ... 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 Nk represent the set of integers with precisely k prime factors (counted with multiplicity), then we have f(N1) > f(N2) > f(N3) > · · ·. 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. [ABSTRACT FROM AUTHOR]