Improvements of Dusart's lower bound for $ e^\gamma \log x \prod_{p\le x} \frac{p-1}{p}$.

Question

Let $\gamma$ be the Euler-Mascheroni constant. In this paper (Theorem 6.12) it is proved that for $x\ge 2793$, $$ e^\gamma \log x \prod_{p\le x} \frac{p-1}{p}> 1-\frac{1}{5 \left(\log x\right)^2+1}. $$

I've been wondering, is there a $C$ such that for $x>C$, $$ e^\gamma \log x \prod_{p\le x} \frac{p-1}{p}>1-\frac{1}{x^2} \space?$$

Answer 1

No. Diamond and Pintz proved that for some constant $c>0$, there are arbitrarily large values of $x$ for which $$ e^\gamma \log x \prod_{p\le x} \frac{p-1}p < 1 - \frac{c\log\log\log x}{x^{1/2}\log x}. $$ Lamzouri conjectures that the right-hand side can be improved to $$1 - \frac{(\log\log\log x)^2}{(2\pi+\varepsilon) x^{1/2}\log x}$$ but no further.