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 recent 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 Gál, the method of Aistleitner, Berkes, and Seip, and a certain completeness property of extremal sets of square-free numbers.

Final version, to appear in Bulletin of the London Mathematical Society

Cited by in corpus (5)

• Large GCD sums and extreme values of the Riemann zeta function
• Additive energy and the metric Poissonian property
• A metric theory of minimal gaps
• Gál-type GCD sums beyond the critical line
• Lower bounds for the maximum of the Riemann zeta function along vertical lines