TITLE: Has the arcsine law been generalised to higher order divisor functions?
QUESTION [11 upvotes]: The arcsine law for the distribution of the logarithms of the divisors of an integer $n$ states that 
$$
\frac{1}{x}\sum_{n\leq x}\frac{1}{d(n)}\sum_{\substack{q|n\\q\leq n^{A}}}1\sim \frac{2}{\pi}\arcsin \sqrt A
$$
for $0<A\leq 1$. This was proved by Deshouillers, Dress and Tenenbaum (Acta Arithmetica (1979)
Volume 34, Issue 4, 273-285). More generally, I would like to know if the mean value of
$$
\frac{1}{d_k(n)} \sum_{\substack{q|n\\q\leq n^{A}}}d_{k-1}(q)
$$ 
for $k>1$ has also appeared in the literature, or if this is known but perhaps unpublished?

REPLY [3 votes]: There's an extension of the Deshouillers-Dress-Tenenbaum theorem by Bareikis and Manstavičius, http://doai.io/10.4064/aa126-2-5, which almost treats the question you ask. I'm not sure why the condition $f(p^\ell) \ll 1$ is imposed there; as far as the method goes the condition $f(p^\ell) \ll \ell^C$ ($C$ absolute) should be fine.