A conjecture of R. R. Hall on Farey points (Q2720960)

Let NEWLINE\[NEWLINE\gamma_1= \frac{1}{Q}< \gamma_2<\cdots< \gamma_{N-1}\leq \frac{Q-1}{Q}< \gamma_N= 1NEWLINE\]NEWLINE denote the Farey sequence of order \(Q\). Put \(\gamma_{N+j}= \gamma_j+1\) for \(0\leq j< N\). A problem is the asymptotic behaviour of NEWLINE\[NEWLINES(Q)= \sum_{j=1}^N (\gamma_{j+h}- \gamma_j)^2NEWLINE\]NEWLINE as \(Q\to\infty\). A conjecture of \textit{R. R. Hall} [Acta Arith. 66, 1-9 (1994; Zbl 0831.11022)] is that NEWLINE\[NEWLINES(Q)= \frac{c(h)\log Q} {Q^2}+ \frac{d(h)} {Q^2}+ o \biggl( \frac{1} {Q^2} \biggr)NEWLINE\]NEWLINE holds for any \(h\geq 1\), where \(c(h)\), \(d(h)\) are constants depending only on \(h\). R. R. Hall could prove this conjecture only for \(h=1,2\). The main result of this paper is a proof of this asymptotic representation for any \(h\geq 3\). Moreover, a better error term is given.