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