# Uniform lower bound for $|\zeta(1 + it)|$

With $$\zeta$$ denoting the Riemann zeta function, prove that

$$|\zeta(1+it)| \gg \frac{1}{\log^7 t}$$

uniformly for all $$t\geq 10$$.

As a hint, it is suggested to use a part of the proof that $$\zeta$$ has no roots on the line $$\Re(s)=1$$.

I guess they refer to the inequality $$\zeta(\sigma)^3|\zeta(\sigma + it)|^4|\zeta(\sigma + 2it)| \geq 1$$

Also, if you want, you can use the bounds $$|\zeta(\sigma + it)| = O(\log t)$$ and $$|\zeta'(\sigma + it)| = O(\log^2 t)$$ uniformly on all $$1\leq \sigma \leq 2$$, $$t \geq 10$$ (though I am pretty sure they work for $$t\geq \frac{5}{2}$$, as well!).

My thought was to (if necessary) replace $$t$$ by $$2t$$ in the above inequality to get a new one, and then multiply both to get big powers and use the $$\zeta$$ estimate above, but in any case I do not know how to get rid of $$\zeta(\sigma))$$ which is problematic since the argument inside has imaginary part $$0$$ and $$\sigma =1$$ is a simple pole. Perhaps an inequality like $$\zeta(\sigma) \leq \frac{\sigma}{\sigma-1}$$ might help?! No idea.

Any help appreciated!

From the inequalities you posted we have $$\left|\frac{1}{\zeta\left(\sigma+it\right)}\right|\leq\zeta\left(\sigma\right)^{3/4}\left|\zeta\left(\sigma+2it\right)\right|^{1/4}\ll\frac{\log^{1/4}\left(t\right)}{\left(\sigma-1\right)^{3/4}},\,\sigma>1$$ and $$\zeta\left(1+it\right)-\zeta\left(\sigma+it\right)=-\int_{1}^{\sigma}\zeta^{\prime}\left(s+it\right)ds\ll\left(\sigma-1\right)\log^{2}\left(t\right),\,\sigma>1-A/\log\left(t\right),\,A>0$$ hence $$\left|\zeta\left(1+it\right)\right|\gg\frac{\left(\sigma-1\right)^{3/4}}{\log^{1/4}\left(t\right)}-\left(\sigma-1\right)\log^{2}\left(t\right),\sigma>1-A/\log\left(t\right),\,A>0$$ then tacking $$\sigma-1=A_{1}\log^{-9}\left(t\right)$$ where $$A_{1}>0$$ is sufficiently small we get $$\left|\zeta\left(1+it\right)\right|\gg\frac{1}{\log^{7}\left(t\right)}$$ since all terms have the same order. This proof is taken from Titchmarsh, the theory of the Riemann Zeta function, p. $$50-51$$.