I wondered if it is possible to refute or prove this conjecture that I've considered.
Conjecture. There exists a positive real number $x_0$ such that $$\frac{x}{\varphi(x)}<\frac{\zeta(2)\zeta(3)}{\zeta(6)}\cdot W_0\left(\exp\left(-W_{-1}\left(-\frac{1}{\log x}\right)\right)\right)\tag{1}$$ holds for all integer $x>x_0$, where $\varphi(x)$ denotes the Euler's totient function, $W_k(x)$ the analytic continuation, the different branchs, of the Lambert $W$ function and $\zeta(s)$ the Riemann's zeta function.
The Wikipedia has an article for Lambert $W$ function and other for Euler's totient function.
Question. Is it possible to prove previous conjecture? For what choice of $x_0$? Many thanks.
I doubt that the known inequalities that involve the Euler's totient function are enough to prove it.
I tried to create it with the purpose that the RHS of the inequality $(1)$ will be approximately a multiple of $\log\log x$, but I have no a good motivation for the choice of the value of the constant $\frac{\zeta(2)\zeta(3)}{\zeta(6)}$ in the RHS.