Let $\omega(n)$ the number of distinct prime factors of $n$ (counted without multiplicity). A famous consequence of Turán–Kubilius inequality is $$ \sum_{n\leq x}(\omega(n)-\log\log x)^2=O(x\log \log x). $$
However, I am interested in a small variation of the previous result. Indeed, I would like to obtain a similar result to $$ \sum_{n\leq x}(\omega(n)-A\log\log x)^2, $$ for some previously fixed $A\in (0,1)$ (e.g., $A=1/4$ is enough for me).
Any suggestion? Thanks in advance.
For any real $A \neq 1$ we have $$ \sum_{n \leq x} (\omega(n) - A \log \log x)^2 \sim (1-A)^2 x (\log \log x)^2 $$ as $x \to \infty$. This is a consequence of the quoted result $$ \sum_{n \leq x} (\omega(n) - \log \log x)^2 \ll x \log \log x, $$ via the triangle inequality: the vector $(\omega(n))_{n \leq x}$ of length $\lfloor x \rfloor$ is within $O( (x\log\log x)^{1/2})$ of $(\log \log x) \cdot {\bf 1}$, and that vector has norm $\lfloor x \rfloor^{1/2} \log\log x$, so its distance from $A (\log \log x) \cdot {\bf 1}$ is $|1-A| \lfloor x \rfloor^{1/2} \log\log x$, which grows faster than $(x\log\log x)^{1/2}$ for any $A \neq 1$.
Intuitively, "$\sum_{n \leq x} (\omega(n) - \log \log x)^2 \ll x \log \log x$", or even "$\sum_{n \leq x} (\omega(n) - \log \log x)^2 = o(x (\log \log x)^2)$", says that the typical $n \leq x$ has $\omega(n) = (1+o(1)) \log \log x$. That means that typically $\omega(n)$ differs from $A \log \log x$ by $(1-A+o(1)) \log \log x$, so the sum of the squares of the differences grows as $(1-A)^2 x (\log \log x)^2$.
Related:
R.L. Duncan, Some Applications of the Turan-Kubilius Inequality in Proceedings of the American Mathematical Society 30(1), September 1971, has proved the following:
Let $g:\mathbb{N}\rightarrow \mathbb{R}$, with $g(0)=0,g(1)\neq 1,$ and define $f(n)=\sum_{i=1}^rg(a_i)$ where $n=p_1^{a_1} \cdots p_r^{a_r}.$
If $g(n)=O(b^{n/2}),0<b<2,$ then $$ \sum_{n\leq x} (f(n)-g(1) \log\log x)^2 \leq c x \log\log x. $$
My source is Sandor and Mitrinovic, Handbook of Number Theory, Vol. I, Chapter V. but you can also see the first page of the paper on Jstor where the result is stated.
