Theorem 12 Let $x \geqslant 2$. We have
$$\prod_{p \leqslant x}\left(1-\frac{1}{p}\right)=\frac{\mathrm{e}^{-A_{1}}}{\ln x}+O\left(\frac{1}{\ln ^{2} x}\right)$$

where $A_{2}=A_{1}-A_{3}, A_{1}$ is given by equation (45),
$$A_{3}=\sum_{p}\left[\ln \left(1-\frac{1}{p}\right)+\frac{1}{p}\right]$$

Using
$$\sum_{p \leqslant x}\left(\ln \left(1-\frac{1}{p}\right)+\frac{1}{p}\right)=A_{3}+O\left(\frac{1}{x}\right)$$