The index of a Farey sequence. (Q1396353)

Let \(F_N= \{x_i,\,i= 1,2,\dots, R\}\) denote the Farey sequence of order \(N\), such that \(x_1= 1/N\) and \(x_R= 1\). Define \(x_{i+R}= x_i+ 1\) for all \(i\). Let \(x_{i-1}={a\over r}\), \(x_i= {b\over s}\), \(x_{i+1}= {c\over t}\), then \(\nu(x_i)= {r+ t\over s}= {a+c\over b}\) is called the index of the fraction \(x_i\). The authors prove asymptotic expansions for the mean values of \(\nu(x_i)\), \(\nu^2(x_i)\) and some related arithmetic functions.