Fix a finite collection of distinct prime numbers $(p_1, p_2, \dots, p_s)$, denote their product by $N$. For a natural number $n$ let $\beta(n)$ be the number of $k$, $k\leq n$, for which $k$ and $N$ are relatively prime. Consider the quantity $$ K(p_1, p_2, \dots, p_s)=\sum_{n\geq 1}\left(\frac{\beta(n)}{n}-\theta\right)^2, $$ where $$ \theta=\frac{\beta(N)}{N}=\prod_{i=1,\dots, s}\left(1-\frac{1}{p_i}\right)=\lim_{n\to\infty}\frac{\beta(n)}{n}. $$
Now consider the natural partial order (by inclusion) on the set of all finite collections of prime numbers. It gives us a net. My question is: does $K(p_1, p_2, \dots, p_s)$ tend to infinity along this net?
