Refinements of Gál's theorem and applications (Q340400)

Let \((n_k)_{1\leq k \leq N}\) be a sequence of distinct positive integers. This paper addresses the problem of bounding greatest common divisor (GCD) sums of the form NEWLINE\[NEWLINE \frac{1}{N} \sum_{k, \ell=1}^{N} \frac{(\mathrm{gcd}(n_k,n_{\ell}) )^{2\alpha}}{(n_k n_{\ell})^{\alpha}},NEWLINE\]NEWLINE for \(0< \alpha \leq 1\). \textit{I. S. Gál} solved the problem for \(\alpha=1\) in 1949 [Nieuw Arch. Wiskd., II. Ser. 23, 13--38 (1949; Zbl 0031.25601)] . He showed that the GCD sum is bounded by \(C (\log \log N)^2\) for an absolute constant \(C>0\) and that this bound is optimal up to the constant. In the present paper, the authors give a new (and short) proof of the Theorem of Gál and determine this optimal constant \(C\) as \(N \rightarrow \infty\). Moreover, the authors find a transparent explanation of the relationship between the maximal size of the Riemann zeta function on vertical lines and bounds on GCD sums. In addition, they rederive and improve various recent results of \textit{C. Aistleitner} et al. [J. Eur. Math. Soc. (JEMS) 17, No. 6, 1517--1546 (2015; Zbl 1344.11053)] and settle different open questions from this paper as well as from the related paper of \textit{C. Aistleitner} et al. [Acta Arith. 168, No. 3, 221--246 (2015; Zbl 1339.42008)].