# An estimate for relatively prime numbers

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?

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 interesting. But it seems it can be stated more simply! – user59671 Feb 12 at 2:55