I'm reading through an article on Paul Erdos (http://www.ams.org/notices/199801/vertesi.pdf) and on page 22 they mention the following:
Calling 714 and 715 a “Ruth-Aaron pair”, we conjectured that such pairs have density 0: that is, the set of n, such that the sum of the prime factors of n is equal to the sum of the prime factors of n+ 1, has density 0.
What does it mean for pairs to have a density of zero? I've searched around the webs and it seems to be in general usage without much being said about it otherwise? I get that they are defining it by saying "the set of n...."; but, I'm still not sure what it means.
Let $A(x)$ be the number of integers $n$, $n\le x$, such that $n$ and $n+1$ form a Ruth-Aaron pair. What "density zero" means is that $$\lim_{x\to\infty}{A(x)\over x}=0$$