Conjecture(1)
Merten's third theorem says:
$$\lim_{L\to\infty}\ln L\prod_{p\le L}\left(1-\frac1p\right)=e^{-\gamma}$$
we have a wild discussion here around the table whether it is possible to deduct from this theorem that:
over an $L\in \Bbb N$ the probability $\Psi(x)$ to hit the next integer to $x\in \Bbb R$ which is not divisible by primes $p\le L$, would be decreasing when $x$ increasing according to
$$\Psi(x) \sim \frac {e^{\gamma}\ln L}{\ln x}$$
We can not find a consensus. May we get your help. Is this correct? Why, what is the formal relation?
(1) Conjecture to be cited Vaseghi 2013