Lower Bounds for the Riemann Zeta Function on the Critical Line

We establish a relation between the lower bound for the maximum of the modulus of $\zeta(1/2+iT+s)$ in the disk $|s|\le H$ and the lower bound for the maximum of the modulus of $\zeta(1/2+iT+it)$ on the closed interval $|t|\le H$ for $0<H(T)\le1/2$. We prove a theorem on the lower bound for the maximum of the modulus of $0<H(T)\le1/2$ on the closed interval $|t|\le H$ for $40\le H(T)\le\log\log T$.