Gál-type GCD sums beyond the critical line

Abstract

We prove that \[ \sum_{k,{\ell}=1}^N\frac{(n_k,n_{\ell})^{2α}}{(n_k n_{\ell})^α} \ll N^{2-2α} (\log N)^{b(α)} \] holds for arbitrary integers $1\le n_1<\cdots <n_N$ and $0<\alpha <1/2$ and show by an example that this bound is optimal, up to the precise value of the exponent $b(\alpha )$. This estimate complements recent results for $1/2\le \alpha \le 1$ and shows that there is no "trace" of the functional equation for the Riemann zeta function in estimates for such GCD sums when $0<\alpha <1/2$.

A few minor misprints have been removed

References in corpus (1)

• GCD sums and complete sets of square-free numbers

Cited by in corpus (1)

• Decoupling theorems for the Duffin-Schaeffer problem