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1$ has also appeared in the literature, or if this is known but perhaps unpublished?



Has the arcsine law been generalised to higher order divisor functions?
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.


