Three-four-one zero-free-region argument
= Three-four-one zero-free-region argument
For $F=-\zeta'/\zeta$ and $\sigma>1$, positivity of
$$
3+4\cos\theta+\cos2\theta=2(1+\cos\theta)^2
$$
term by term in the <Euler product> gives
$$
3F(\sigma)+4\Re F(\sigma+it)+\Re F(\sigma+2it)\geq0.
$$
Comparing this with the pole of $F$ at one and the local poles of $\zeta'/\zeta$ at zeros proves the classical zero-free region.