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In this paper, we obtain a series of new conditional lower bounds for themodulus and the argument of the Riemann zeta function on very short segments ofthe critical line, based on the Riemann hypothesis. In particular, theconditional solution of one problem of A.A.Karatsuba is given. Some typos ofthe previous versions are corrected (in particular, the important remark ofProf. Yan Fyodorov is taken into account). The reference to the grant ofRussian Scientific Fund is also added.

On the large values of the Riemann zeta-function on short segments of the critical line