Average value of the prime omega function $\Omega$ on predecessors of prime powers For a positive integer $n$, the prime omega function value $\Omega(n):=\sum_{p\mid n}{\nu_p(n)}$ counts the number of prime divisors of $n$ with multiplicities. A result of Hardy and Wright, [1, Theorem 430 on p. 472], implies that $\frac{1}{x}\sum_{n\leq x}{\Omega(n)}\sim\log\log{x}$ as $x\to\infty$.
Question:
Letting the variable $q$ range over prime powers, is it true that
$\left(\sum_{q\leq x}{1}\right)^{-1}\cdot\sum_{q\leq x}{\Omega(q-1)}\sim\log\log{x}$
as $x\to\infty$?
This question is motivated by some work in progress concerning certain algorithms over finite fields $\mathbb{F}_q$. An affirmative answer would imply that these algorithms are efficient for "most" finite fields. In fact, it would suffice if
$\left(\sum_{q\leq x}{1}\right)^{-1}\sum_{q\leq x}{\operatorname{mpe}(q-1)}\in O(\log\log{x})$
where $\operatorname{mpe}(n):=\max_{p\mid n}{\nu_p(n)}\leq\Omega(n)$. The following table provides some computational evidence, namely the values of $f(x):=\left(\log\log{x}\sum_{q\leq x}{1}\right)^{-1}\sum_{q\leq x}{\Omega(q-1)}$ for $x=10^n$ with $n\in\{5,6,7,8,9\}$.




$x$
$10^5$
$10^6$
$10^7$
$10^8$
$10^9$




$f(x)$
1.91446
1.86387
1.82433
1.7924
1.76574




Reference:
[1] G.H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers. Edited and revised by D.R. Heath-Brown and J.H. Silverman. With a foreword by Andrew Wiles, Oxford University Press, Oxford, 6th edn. 2008.
 A: Yes, this is true.
First, let us observe that replacing prime powers with primes cannot make a difference, and the same goes to replacing $\Omega$ with $\omega$.
H. Halberstam proved that
$$\frac{\sum_{p \le x} \omega(p-1)}{\sum_{p \le x}1}\sim \log \log x,$$
in "On the distribution of additive number-theoretic functions. III.", J. London Math. Soc. 31 (1956), 14–27. See Theorem 1 of his paper.
In fact, he proved his result for $p-1$ replaced by $f(p)$ for any irreducible polynomial $f$ with integer coefficients.
In fact, a central limit theorem can be proved for $\omega(p-1)$, and this is due to Barban (according to Elliott's book mentioned below). Some modern references for this:

*

*Peter D. T. A. Elliott, "Probabilistic number theory. II", Mathematischen Wissenschaften 240. Springer-Verlag, Berlin-New York, 1980.

*Krishnaswami Alladi, "Moments of additive functions and the sequence of shifted primes", Pacific J. Math. 118 (1985), no. 2, 261–275.

*Andrew Granville and Kannan Soundararajan, "Sieving and the Erdős-Kac theorem", Equidistribution in number theory, an introduction, 15–27, NATO Sci. Ser. II Math. Phys. Chem., 237, Springer, Dordrecht, 2007.

A: One can prove that in fact the function mpe has bounded average which clearly improves $\log \log$. If a positive integer n has $mpe(n)=m$ then there exists a prime power p^m that divides n hence the cardinality of such integers below x will be at most x/p^m. Summing over all p will give cardinality O(x/2^m). Hence the average of mpe(n) over all integers n\leq x will be O(xS) where S is the concergent sum from m=1 to infinity of m/2^m. The same proof easily works for q-1 if one uses Brun-Titchmarch when p^m goes up to sqrt x
and the trivial upper bound for larger p^m. In fact one can push these arguments a little to prove that the average of mpe exists.
