An estimate for relatively prime numbers

Well, this is a 'possibly open problem'. What you are asking basically gives a reformulation of Riemann Hypothesis. Note that, $\frac{\beta (n)}{n} = \sum_{d|N}\mu(d)[\frac{n}{d}]$. If you are familiar with Legendre Sieve you may note that $K(p_1, p_2, \dots, p_s)=\sum_{n\geq 1}\left(\frac{\beta(n)}{n}-\theta\right)^2 = \sum_{n\geq 1}\frac{\left(\sum_{d|N}\mu(d)\left\{\frac{n}{d}\right\}\right)^2}{n^2}$. If you use formal sieve method you may be able to get a 'good bound' of the function $K(p_1,...,p_s)$. See the notes. I would like to suggest a different procedure. Introduce a Weighted $l^2$ space where $||x||=\sum_{n=1}^{\infty}\frac{x(n)^2}{n^2}$ whenever convergent. Consider the Hilbert space generated by $\left\{\gamma_n|\gamma_n(k)=\left\{\frac{k}{n}\right\},k=1,2,...\right\}$(See the paper Bagchi's Reformulation which describes the methodology in detail). Consider the vector $x_N=\sum_{d|N}\mu(d)\gamma_d$, and therefore $K(p_1,...,p_s)$ is nothing but $||x_N||^2$. From this one can have a conjecture that RH is true iff $K(p_1,...,p_s)\to 1$ as $N\to\infty$. Hope this helps.