Extreme Values of the Riemann Zeta Function on the 1-Line

Abstract

We prove that there are arbitrarily large values of t such that {equation presented}. 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.