How are the ratios of successive values of the divisor function distributed? 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.
 A: Under GRH, Titchmarsh showed 1931 that $\sum_{p\leq x}\tau(p+a)\sim C(a)x$, where summation runs over primes only. 1963 Linnik proved the same unconditionally, hence there are many $n$ such that $d(n)=2$, $d(n+1)\gg\log n$ and similarly for $n-1$. Hence $r(n)$ is  quite often as big as $\log n$ and as small as $\frac{1}{\log n}$. 
In general $r(n)$ is much closer to 1. Using some standard techniques in probabilistic number theory you can show that the joint distribution of $(\omega(n), \omega(n+1))$ is normal with mean and variance $\log\log x$, hence the distribution of $\log r(n)$ is normal with variance $2\log\log n$. 
You can construct extreme values of $r(n)$ by taking an integer $n$ with many divisors, and look for a prime $p\equiv \pm 1\pmod{n}$. By Heath-Browns version of Linnik's theorem, for $n>n_0$ such a prime $p$ exists satisfying $p\leq n^{5.5}$, hence $\max_{p\leq x} d(p+1)\geq \max_{n\leq x^{2/11}}d(n)$, thus for some small $c>0$ we have that $r(n)>e^{c\log n/\log\log n}$ has infinitely many solutions.
