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1$ has also appeared in the literature, or if this is known but perhaps unpublished?
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2 Answers
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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.
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$\begingroup$ Thank you Sary - this is even more general. $\endgroup$ Commented Sep 25, 2018 at 18:28
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Well here's the answer for anyone(else) who's interested. We have
$$
\frac{1}{x}\sum_{n\leq x}\left(\frac{1}{d_k(n)}\sum_{\substack{q|n\\q\leq n^A}}
d_{k-1}(q)\right)\sim \frac{\sin\pi/k}{\pi}\int_{0}^{A}t^{-\frac{1}{k}}(1-t)^{\frac{1}{k}-1}dt,$$
which is the generalised arcsine distribution function, in other words, the $B(1/k,1-1/k)$ distribution.
I'm not inclined to write out the proof in details right now (especially because I asked the question), but it follows along elementary lines using any of a number of theorems on mean values of multiplicative functions (of the Selberg-Delange type for example), partial summation and some basic combinatorics.