= Solution
Differentiate the <locally uniformly convergent> logarithm of the <Euler product>. Its <logarithmic derivative> is
$$
-\frac{\zeta'(s)}{\zeta(s)}=\sum_p\sum_{m\ge1}\frac{\log p}{p^{ms}}=\sum_{n\ge1}\frac{\Lambda(n)}{n^s},\qquad\Re s>1.
$$
The differentiated sum is <absolutely convergent>, since $\Lambda(n)\le\log n$. Taking the real parts at the three heights gives
$$
-3\frac{\zeta'(\sigma)}{\zeta(\sigma)}-4\Re\frac{\zeta'(\sigma+it)}{\zeta(\sigma+it)}-\Re\frac{\zeta'(\sigma+2it)}{\zeta(\sigma+2it)}=\sum_{n\ge1}\frac{\Lambda(n)}{n^\sigma}\bigl(3+4\cos(t\log n)+\cos(2t\log n)\bigr)\ge0.
$$
Each summand is nonnegative because its bracket is $2(1+\cos(t\log n))^2$. This proves the derivative form of the <three-four-one zero-free-region argument>.
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