GCD sums and complete sets of square-free numbers

It is proved that $$\sum _{k,\mathrm{\ell }=1}^N\frac{\gcd (n_k,n_{\mathrm{\ell }})}{\sqrt{n_k{n}_{\mathrm{\ell }}}}\ll N\exp \left(C\sqrt{\frac{\log N\log \log \log N}{\log \log N}}\right)$$ holds for arbitrary integers $1\le n_1<\cdots <n_N$. This bound is essentially better than that found in a recent paper of Aistleitner, Berkes, and Seip and can not be improved by more than possibly a power of $1\mathrm{/}\sqrt{\log \log \log N}$. The proof relies on ideas from classical work of G'{a}l, the method of Aistleitner, Berkes, and Seip, and a certain completeness property of extremal sets of square-free numbers.