Equivalences involving the Farey series

Equidistribution of Farey sequence: Let $r_v$ be the elements of the Farey sequence of order $N$, $v=1,2,\dots \Phi(N)$ where $\Phi(N)=\sum_{n=1}^N\phi(n)$. Let $\delta_v=r_v-v/\Phi(N)$. Then RH if and only if

\begin{displaymath}\sum_{v=1}^{\Phi(N)} \delta_v^2\ll N^{-1+\epsilon}.\end{displaymath}

Also, RH if and only if

\begin{displaymath}\sum_{v=1}^{\Phi(N)} \vert\delta_v\vert\ll N^{1/2+\epsilon}.\end{displaymath}

Here is [a good bibliography] on this subject.

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