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On large values of the Riemann zeta-function on short segments of the critical line. (English) Zbl 1377.11093
Summary: We obtain a series of new conditional lower bounds for the modulus and the argument of the Riemann zeta function on very short segments of the critical line, based on the Riemann hypothesis. In particular, we prove that for any large fixed constant $$A>1$$ there exist (non-effective) constants $$T_{0}(A)>0$$ and $$c_{0}(A)>0$$ such that the maximum of $$|\zeta (0.5+it)|$$ on the interval $$(T-h,T+h)$$ is greater than $$A$$ for any $$T>T_{0}$$ and $$h = (1/\pi)\ln\ln\ln{T}+c_{0}$$.

##### MSC:
 11M06 $$\zeta (s)$$ and $$L(s, \chi)$$ 11M26 Nonreal zeros of $$\zeta (s)$$ and $$L(s, \chi)$$; Riemann and other hypotheses
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