Farey sequences
The Farey sequence \((h_1(n),\dots, h_{\Phi(n)} (n))\) consists of the members of \(H_n= \{\frac ab: a,b\in \mathbb{N}\), \(1\leq a\leq b\), \((a,b)=1\}\) arranged in increasing order. Here, \(\Phi(n)= | H_n| = \sum_{m\leq n}\varphi(m)\), where \(\varphi\) is Euler's function. For fixed \(n\), let \(\delta_i= \delta_i(n)= h_i(n)- \frac{i}{\Phi(n)}\); define \(S_p(n)= (\sum_i| \delta_i^p|)^{1/p}\) and \(\Delta(n)= \max\{| \sum_{i=1}^m \delta_i|: 1\leq m\leq \Phi(n)\}\). Both \(S_p(n)\) and \(\Delta(n)\) are related to the Riemannian hypothesis. The author gives various estimates, in particular lower bounds, for both \(S_p(n)\) and \(\Delta(n)\), some of them depending on \(\theta= \sup\{\sigma: \xi(\sigma+i_t)=0\}\). One of his bounds, namely \(S_2(n)\geq Cn^{-1/2} (\log n)^{1/2}\) shows the falsity of a conjecture of Sato, that \(S_2(n)= O(n^{-1/2}\log \log \log n)\).
scientific article; zbMATH DE number 1237982
scientific article