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

Let alpha is an element of(1/ 2, 1) be fixed. We prove that max(0 <= t <= T) vertical bar zeta(alpha + it)vertical bar >= exp(c(alpha)(log T)(1-alpha)/(log log T)(alpha)) for all sufficiently large T, wherewe can choose c(alpha) = 0.18(2 alpha-1)(1-alpha). The same result has already been obtained by Montgomery, with a smaller value for c(alpha). 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 is an element of[0, T] for which vertical bar zeta(alpha + it)vertical bar is of the order mentioned above.