Lower bounds for the maximum of the Riemann zeta function along vertical lines

Abstract

Let α ∈ (1/2, 1) be fixed. We prove that (Formula presented.) for all sufficiently large T, where we can choose cα = 0.18(2α - 1)1-α. The same result has already been obtained by Montgomery, with a smaller value for cα. However, our proof, which uses a modified version of Soundararajan’s “resonance method” together with ideas of Hilberdink, is completely different from Montgomery’s. This new proof also allows us to obtain lower bounds for the measure of those t ∈ [0, T] for which |ζ(α + it)| is of the order mentioned above.