= Weak logarithmic zero-free region for the Riemann zeta function
{title2=$|1/\zeta(\sigma+it)|\ll\log^7(|t|+2),\quad\sigma\ge1-c/\log^9(|t|+2)$}
The logarithm of the <Euler product> and $3+4\cos\theta+\cos2\theta\ge0$ give $\zeta(\sigma)^3|\zeta(\sigma+it)|^4|\zeta(\sigma+2it)|\ge1$. At $\sigma=1+a/\log^9|t|$, upper bounds $\zeta(\sigma)\ll\log^9|t|/a$ and $|\zeta(\sigma+2it)|\ll\log|t|$ imply the lower bound $|\zeta|\gg a^{3/4}\log^{-7}|t|$. The <Cauchy estimate for derivatives> gives $|\zeta'|\ll\log^2|t|$ nearby, allowing a sufficiently small leftward displacement of order $\log^{-9}|t|$. This elementary region is weaker than the classical logarithmic region but gives an explicit reciprocal bound by simple estimates.
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