= Three-four-one product proof of zeta boundary nonvanishing
{title2=$\zeta(\sigma)^3|\zeta(\sigma+it)|^4|\zeta(\sigma+2it)|\geq1$}
For $\sigma>1$, the logarithm of the displayed product is a prime-power sum with coefficients $3+4\cos\theta+\cos2\theta=2(1+\cos\theta)^2\geq0$. A zero of order $a\geq1$ at $1+it$, with $t\neq0$, 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.
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