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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.

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