Large greatest common divisor sums and extreme values of the Riemann zeta function

It is shown that the maximum of |ζ(1/2+it)| on the interval T1/2≤t≤T is at least exp((1/2+o(1))logTlogloglogT/loglogT). Our proof uses Soundararajan’s resonance method and a certain large greatest common divisor sum. The method of proof shows that the absolute constant A in the inequality sup 1≤n1<⋯<nN∑k,ℓ=1Ngcd(nk,nℓ)nknℓ≪Nexp(AlogNlogloglogNloglogN), established in a recent paper of ours, cannot be taken smaller than 1.