On the asymptotic behaviour of Franel's sum and the Riemann hypothesis (Q1193238)

Let \(\varphi(n)\) and \(\mu(n)\) be the functions of Euler and Möbius, respectively, and write \(\Phi(N)=\sum_{n\leq N}\varphi(n)\) and \(M(N)=\sum_{n\leq N}\mu(n)\). The Farey sequence of order \(N\) is the increasing sequence of fractions \(h/k\), where \(h\), \(k\) are coprime integers satisfying \(1\leq h\leq k\leq N\), and there are \(\Phi(N)\) such fractions. In 1912 \textit{J. E. Littlewood} [C. R. Acad. Sci. 154, 263-266 (1912)] showed that the estimate \(M(N)\ll N^{{1\over 2}+\varepsilon}\), for every \(\varepsilon>0\), is equivalent to the Riemann hypothesis. In 1924, \textit{J. Franel} [Göttinger Nachr., 198-201 (1924; JFM 50.0119.01)] studied the variance \[ Q(N)=\sum_{m\leq\Phi(N)}\left(\rho(m,N)- {m\over{\Phi(N)}}\right)^ 2, \] where \(\rho(m,N)\) is the \(m\)th term of the Farey sequence of order \(N\), and he showed that the estimate \(Q(N)\ll N^{-1+\varepsilon}\) is also equivalent to the Riemann hypothesis. The authors prove the following. Let \(g(x)\) be a positive function with a nonnegative derivative, and suppose that there exists \(\theta>-{1\over 2}\) such that \(g(x)x^ \theta\) is decreasing for large \(x\). Then the two estimates \(M(N)=O(N/g(N))\) and \(Q(N)=O((1/g^ 2(N))\) are equivalent, and that the same is also true when \(O\) here is replaced by \(o\). From numerical computations of \(Q(N)\) for \(N\leq 10000\), the authors also speculate that \(NQ(N)/\log\log N\) may have a positive limit as \(N\to\infty\).