Gál-type GCD sums beyond the critical line

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 $0<α<1/2$ and show by an example that this bound is optimal, up to the precise value of the exponent $b(α)$. This estimate complements recent results for $1/2\le α\le 1$ and shows that there is no "trace" of the functional equation for the Riemann zeta function in estimates for such GCD sums when $0<α<1/2$.

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