GCD sums and complete sets of square-free numbers

Abstract: It is proved that [ \sum_{k,{\ell}=1}^N\frac{\gcd(n_k,n_{\ell})}{\sqrt{n_k n_{\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 paper of Aistleitner, Berkes, and Seip and can not be improved by more than possibly a power of $1/\sqrt{\log\log\log N}$. 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.