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arXiv · 1512.03758

Gál-type GCD sums beyond the critical line

Andriy Bondarenko·Titus Hilberdink·Kristian Seip
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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 $0<α<1/2$ 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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Andriy Bondarenko, Titus Hilberdink, Kristian Seip. 2016-04-11. Gál-type GCD sums beyond the critical line. https://doi.org/10.1016/j.jnt.2016.02.017

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