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Gál-type GCD sums beyond the critical line (1512.03758v2)
Published 11 Dec 2015 in math.NT
Abstract: 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 $0<\alpha<1/2$.