On the small values of the Riemann zeta-function on the critical line

The main goal of the talk is to describe some recent results of the speaker concerning the existence of very small values of the Riemann zeta-function on the sequence of Gram points on the critical line. In particular, the following result will be discussed:
$$ \liminf_{n\to+\infty}\biggl|\zeta\biggl(\frac{1}{2}+it_{n}\biggr)\biggr|e^{A\sqrt{\ln\ln{n}}}=0, $$
where $A$ is an arbitrary large positive number.