Literature on analogous arithmetic function of logarithm function In number theoretical estimations, often we take the logarithms of a natural number to express it properly. A perfect example of this is the von-Mangoldt function. I am looking for an analogous arithmetic function of the logarithm function which is of the form $$\beta(n)=\sum_{p_{i}^{\alpha_{i}}\mid n}\alpha_{i}\cdot \log{\log{p_{i}}}$$ where $n=p_{1}^{\alpha_{1}}\cdots p_{s}^{\alpha_{s}}.$ Is this function well known in literature?
P.S. I am sorry if this post doesn't suit for MathOverflow and would delete it if suggested.
 A: Your function $\beta$ is a completely additive function, in the sense that $\beta(nm)=\beta(n)+\beta(m)$ for all $n,m$.
There is a vast literature on the statistical behavior of additive functions, e.g. Erdös and Kac's 1940 paper and various textbooks: Kubilius' book "Probabilistic methods in the theory of numbers", Elliott's books and (more recently) Tenenbaum's book. And this is just to name a few...
These paper and books focus on the general theory rather than specific examples, and I do not know if your specific example got any attention (is there a reason it should have?). A large portion of the classical literature (e.g. the Erdös-Kac paper and Kubilius' book) seems to focus on strongly additive functions (those that satisfy $\alpha(p^k)=\alpha(p)$), so not capturing your $\beta$.
A good source for the treatment of completely additive functions is Billingsley's 1974 paper "The Probability Theory of Additive Arithmetic Functions".
His Theorem 3.1 applies to your $\beta$ and guarantees that $\beta(n)$ suitably normalized has a gaussian limiting distribution, just as in the classical Erdös-Kac theorem.
In more detail, given a completely additive (real) function $\beta$ let
$$A_n = \sum_{p \le n} \frac{\beta(p)}{p}, \qquad B_n = \sum_{p \le n} \frac{\beta^2(p)}{p}.$$
A sufficient condition for
$$ \frac{\beta(m) - A_n}{B_n}$$
to tend in distribution to standard gaussian distribution, where $m$ is a number chosen uniformly at random from $\{1,2,\ldots,n\}$, is that
$$ \frac{\max_{p \le n}|\beta(p)|}{B_n} \to 0.$$
Verifying this for $\beta(p)=\log \log p$ (or any power of $\log \log p$) is an exercise in Mertens' theorems and partial summation.
