Almost primes and the Banks-Martin conjecture

TL;DR

This paper investigates the behavior of sums over numbers with a fixed number of prime factors, proving they tend to 1 as the number of prime factors increases and refuting a conjecture about their monotonic decrease.

Contribution

It demonstrates that the sum tends to 1 as the number of prime factors grows and disproves the Banks-Martin conjecture of monotonic decrease, identifying a global minimum at k=6.

Findings

Sum tends to 1 as k approaches infinity

Banks-Martin conjecture is false in general

Global minimum of the sum occurs at k=6

Abstract

It has been known since Erdos that the sum of $1/(n\log n)$ 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 $k=6$.