It is shown that the maximum of $\left|\zeta \right(1/2+it\left)\right|$ on the interval $T^{1/2}\le t\le T$ is at least $exp\left(\right(1/\sqrt{2}+o\left(1\right)\left)\sqrt{logTlogloglogT/loglogT}\right)$. 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

$${sup\hspace{0.17em}}_{1\le n_1<\cdots <n_N}{\sum }_{k,\ell =1}^N\frac{gcd(n_k,n_{\ell })}{\sqrt{n_k{n}_{\ell }}}\ll Nexp\left(A\sqrt{\frac{logNlogloglogN}{loglogN}}\right),$$ established in a recent paper of ours, cannot be taken smaller than $1$.