Extreme values of the Riemann zeta function and its argument
Journal article, Peer reviewed
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Original versionMathematische Annalen. 2018, 372 999-2015. 10.1007/s00208-018-1663-2
We combine our version of the resonance method with certain convolution formulas for ζ(s) and logζ(s) . This leads to a new Ω result for |ζ(1/2+it)| : The maximum of |ζ(1/2+it)| on the interval 1≤t≤T is at least exp((1+o(1))logTlogloglogT/loglogT−−−−−−−−−−−−−−−−−−−−−√) . We also obtain conditional results for S(t):=1/π times the argument of ζ(1/2+it) and S1(t):=∫t0S(τ)dτ . On the Riemann hypothesis, the maximum of |S(t)| is at least clogTlogloglogT/loglogT−−−−−−−−−−−−−−−−−−−−−√ and the maximum of S1(t) is at least c1logTlogloglogT/(loglogT)3−−−−−−−−−−−−−−−−−−−−−−−√ on the interval Tβ≤t≤T whenever 0≤β<1 .