Let \(\{x\}\) be the fractional part of the real number \(x\). Let \( {\mathcal F}_Q\) be the Farey series of the order \(Q\geq 1\), and let \(A(Q)=\sum_{n\leq Q}\varphi(n)\), \(\varphi\) Euler’s function. Let \(\alpha>0\) be a real number. Let \[ F_\alpha(a,b,Q)=\sum_{\substack{ i=1,f_i=\frac{a_i}{q_i}\in {\mathcal F}_Q\\f_i\leq \alpha}} \Bigg(\sum_{\substack{ m\leq \frac{Q}{q_i}\\mq_i\equiv b (\bmod a)}} \log m\Bigg) \] and \[ F(\alpha;a,b,Q)=F_\alpha(a,b,Q)-\alpha F_1(a,b,Q)+\frac12\sum_{\substack{ n\leq Q\\n\equiv b (\bmod a)}} \log n +\frac{Q}{\varphi(a)}\sum_{\substack{ m\leq Q\\(m,a)=1}} \frac{\{\alpha m\}-\frac12}{m}. \] Let \(Q>Q_0\). The author proves that the generalized Riemann Hypothesis for all \(L(s,\chi)\) with Dirichlet characters mod \(a\) is equivalent to the statement \[ \int_0^1|F(\alpha;a,b,Q)|^2 d\alpha\ll Q^{1+\varepsilon} \] for any \(\varepsilon>0\) and for any \(b\) satisfying \((a,b)=1\) and \(1\leq b\leq a\), and to the statement \[ \begin{split} \sum_{\substack{ j=1\\f_j\in {\mathcal F}_Q}} \Bigg|\sum_{i=1,f_i=\frac{a_i}{q_i}\in {\mathcal F}_Q}^j\Bigg(\sum_{\substack{ m\leq \frac{Q}{q_i}\\mq_i\equiv b (\bmod a)}} \log m\Bigg)-f_j\sum_{\substack{ i=1,f_i=\frac{a_i}{q_i}\in {\mathcal F}_Q\\f_i\leq \alpha}} ^A\Bigg(\sum_{\substack{ m\leq \frac{Q}{q_i}\\mq_i\equiv b (\bmod a)}} \log m\Bigg)\\ +\frac12 \sum_{\substack{ n\leq Q\\n\equiv b (\bmod a)}} \log n+\frac{Q}{\varphi(a)}\sum_{\substack{ m\leq Q\\(m,a)=1}} \frac{\{f_jm\}-\frac12}{m}\Bigg|^2\ll Q^{3+\varepsilon}\end{split} \] for any \(\varepsilon>0\) and for any \(b\) satisfying \((a,b)=1\) and \(1\leq b\leq a\). Other results on the uniform distribution of the Farey series are obtained.