  We prove that there are arbitrarily large values of $t$ such that
$|\zeta(1+it)| \geq e^{\gamma} (\log_2 t + \log_3 t) + \mathcal{O}(1)$. This
essentially matches the prediction for the optimal lower bound in a conjecture
of Granville and Soundararajan. Our proof uses a new variant of the "long
resonator" method. While earlier implementations of this method crucially
relied on a "sparsification" technique to control the mean-square of the
resonator function, in the present paper we exploit certain self-similarity
properties of a specially designed resonator function.
