Editing Riemann zeta function (section)

===Estimates of the maximum of the modulus of the zeta function===
Let the functions {{math|''F''(''T'';''H'')}} and {{math|''G''(''s''<sub>0</sub>;Δ)}} be defined by the equalities

: <math> F(T;H) = \max_{|t-T|\le H}\left|\zeta\left(\tfrac{1}{2}+it\right)\right|,\qquad G(s_{0};\Delta) = \max_{|s-s_{0}|\le\Delta}|\zeta(s)|. </math>

Here {{mvar|T}} is a sufficiently large positive number, {{math|0 < ''H'' ≪ log log ''T''}}, {{math|''s''<sub>0</sub> {{=}} ''σ''<sub>0</sub> + ''iT''}}, {{math|{{sfrac|1|2}} ≤ ''σ''<sub>0</sub> ≤ 1}}, {{math|0 < Δ < {{sfrac|1|3}}}}. Estimating the values {{mvar|F}} and {{mvar|G}} from below shows, how large (in modulus) values {{math|''ζ''(''s'')}} can take on short intervals of the critical line or in small neighborhoods of points lying in the critical strip {{math|0 ≤ Re(''s'') ≤ 1}}.

The case {{math|''H'' ≫ log log ''T''}} was studied by [[Kanakanahalli Ramachandra]]; the case {{math|Δ > ''c''}}, where {{math|''c''}} is a sufficiently large constant, is trivial.

[[Anatolii Alexeevitch Karatsuba|Anatolii Karatsuba]] proved,<ref>{{cite journal| first=A. A.| last=Karatsuba| title= Lower bounds for the maximum modulus of {{math|''ζ''(''s'')}} in small domains of the critical strip | pages=796–798| journal= Mat. Zametki| volume=70|issue=5| year=2001}}</ref><ref>{{cite journal| first=A. A.| last=Karatsuba| title= Lower bounds for the maximum modulus of the Riemann zeta function on short segments of the critical line| pages=99–104| journal= Izv. Ross. Akad. Nauk, Ser. Mat.| volume=68|issue=8| year=2004| doi=10.1070/IM2004v068n06ABEH000513| bibcode=2004IzMat..68.1157K| s2cid=250796539}}</ref> in particular, that if the values {{mvar|H}} and {{math|Δ}} exceed certain sufficiently small constants, then the estimates

: <math> F(T;H) \ge T^{- c_1},\qquad G(s_0; \Delta) \ge T^{-c_2}, </math>

hold, where {{math|''c''<sub>1</sub>}} and {{math|''c''<sub>2</sub>}} are certain absolute constants.