One motivation for this question is a paper by Erdos and Hall `Values of the divisor function on short intervals', in which the authors obtain the leading asymptotics

$$x(\log x)^{2(\sqrt 2-1)-\epsilon}\leq\sum_{n\leq x}\min{(d(n),d(n+1))}\leq x(\log x)^{2(\sqrt 2-1)}$$ and

$$\sum_{n\leq x}\max{(d(n),d(n+1),...,d(n+k-1)})\sim kx\log x.$$ In particular, it is possible that the answer to this question will lead to refinements of these bounds. It is also interesting simply as a curiosity.

Let $r(n)=d(n+1)/d(n)$. Heath-Brown proved that $r(n)=1$ for infinity many $n$, but I don't know how often this is expected to happen. Recent advances in knowledge of the gaps between primes suggests that small and large values are common.

1. What is known about the distribution of values of $r(n)$?

2. In particular, is it known that $r(n)$ is neither bounded above or below?

EDIT: $r(n)$ is neither bounded above or below - the proof is easy so I suppose that's not news though.