It is shown that the maximum of on the interval is at least . Our proof uses Soundararajan’s resonance method and a certain large greatest common divisor sum. The method of proof shows that the absolute constant in the inequality
established in a recent paper of ours, cannot be taken smaller than .
"Large greatest common divisor sums and extreme values of the Riemann zeta function." Duke Math. J. 166 (9) 1685 - 1701, 15 June 2017. https://doi.org/10.1215/00127094-0000005X