Gál-type GCD sums beyond the critical line

We prove that ∑k,ℓ=1N(nk,nℓ)2α(nknℓ)α≪N2−2α(logN)b(α) holds for arbitrary integers 1≤n1<⋯<nN1≤n1<⋯<nN and 0<α<1/20<α<1/2 and show by an example that this bound is optimal, up to the precise value of the exponent b(α)b(α). This estimate complements recent results for 1/2≤α≤11/2≤α≤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<1/20<α<1/2.