The expression $\lim_{n \to \infty} \ln(n) \cdot \prod_{\text{p prime} \le n} (1 - \frac{1}p)$ seems to converge to 0.5614372$\dots$
Assuming it converges to $\lambda = 0.5614372\dots$
does that imply $\lim_{n \to \infty} \prod_{\text{p prime} \le n} (1 - \frac{1}p) = \frac{\lambda}{\ln n}$
and therefore Euler's totient function can be expressed as: $\varphi(n) = n \cdot \prod_{\text{p prime} \le n} (1 - \frac{1}p) = \frac{\lambda n}{\ln n}$
