Almost primes and the Banks–Martin conjecture

Abstract

It has been known since Erdős that the sum of 1/(nlog⁡n) over numbers n with exactly k prime factors (with repetition) is bounded as k varies. We prove that as k tends to infinity, this sum tends to 1. Banks and Martin have conjectured that these sums decrease monotonically in k, and in earlier papers this has been shown to hold for k up to 3. However, we show that the conjecture is false in general, and in fact a global minimum occurs at k=6.