 Three-four-one product proof of zeta boundary nonvanishing

For $\sigma >1$, the logarithm of the displayed product is a prime-power sum with coefficients $3+4\cos \theta +\cos 2\theta =2(1+\cos \theta {)}^2\ge 0$. A zero of order $a\ge 1$ at $1+it$, with $t\ne 0$, would make the product vanish like $O((\sigma -1{)}^{4a-3})$: the real factor has a pole of order three, and the third factor is bounded. This contradicts its lower bound one. Hence the Riemann zeta function has no zeros on its line of real part one; its pole at one is not a zero.