Assuming Riemanns hypothesis, I would like to obtain an upper bound on
$$\left|\frac{1}{\zeta(\sigma+it)}\right|$$
for large $t$ and fixed $\sigma$. I believe it should be easy to show that it grows slower than any positive power, perhaps by a good application of the conjecture of Lindelof
$$\zeta\left(\frac12+it\right)=o(t^{\epsilon})~~, |t|\to \infty$$
Any tips or tricks to start this computation would be much appreciated
On the Riemann hypothesis, Theorem 14.2 in Titchmarsh states: For $s=\sigma+it$ and $\epsilon>0$, both $$\left|\frac{1}{\zeta(s)}\right|=O\left(\left| t\right|^{\epsilon}\right)$$ $$\left|\zeta(s)\right|=O\left(\left| t\right|^{\epsilon}\right)$$ $\left| t \right| \to \infty$ and $\sigma\geq 1/2$.