  We prove that \[ \sum_{k,{\ell}=1}^N\frac{(n_k,n_{\ell})^{2\alpha}}{(n_k
n_{\ell})^{\alpha}} \ll N^{2-2\alpha} (\log N)^{b(\alpha)} \] 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 $0<\alpha<1/2$.
