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We investigate real values of the Riemann zeta function on the critical line. We show that if Gram's points do not intersect with the ordinates of the nontrivial zeros of the Riemann zeta function then the Riemann zeta function takes arbitrarily small real values on the critical line.

We establish unconditional lower bounds for certain discrete moments of the Riemann zeta-function and its derivatives on the critical line. We use these discrete moments to give unconditional lower bounds for the continuous moments $I_{k,l}\left(T\right)={\int }_0^T{\left|{\zeta }^{\left(l\right)}(\frac{1}{2}+it)\right|}^{2k}dt$, where $l$ is a non-negative integer and $k\ge 1$ a rational number. In particular, these lower bounds are of the expected order of magnitude for $I_{k,l}\left(T\right)$.