  It is shown that the maximum of $|\zeta(1/2+it)|$ on the interval $T^{1/2}\le
t \le T$ is at least $\exp\left((1/\sqrt{2}+o(1)) \sqrt{\log T \log\log\log
T/\log\log T}\right)$. Our proof uses Soundararajan's resonance method and a
certain large GCD sum. The method of proof shows that the absolute constant $A$
in the inequality \[ \sup_{1\le n_1<\cdots < n_N}
\sum_{k,{\ell}=1}^N\frac{\gcd(n_k,n_{\ell})}{\sqrt{n_k n_{\ell}}} \ll N
\exp\left(A\sqrt{\frac{\log N \log\log\log N}{\log\log N}}\right), \]
established in a recent paper of ours, cannot be taken smaller than $1$.
