Farey series and the Riemann hypothesis. II (Q1271923)

[For Part I, cf. \textit{S. Kanemitsu} and \textit{M. Yoshimoto}, Acta Arith. 75, 351-374 (1996; Zbl 0860.11047)] For \(x\geq 1\), let \(\rho_1<\rho_2<\rho_3<\cdots <\rho_{\Phi (x)}\) be the Farey points \(\rho_\nu =\frac{b_\nu}{c_\nu}\), where \((b_\nu, c_\nu)=1\) and \(0<b_\nu \leq c_\nu \leq x\). For any integrable function \(f\) on \([0,1]\), define \[ E_f (x) = \sum^{\Phi (x)}_{\nu=1} f(\rho_\nu) - \Phi (x) \int^1_0 f(t)dt. \] In this paper the author examines the class of functions \(f\) for which the Riemann hypothesis is equivalent to the truth of the estimate \(E_f (x) = O(x^{\frac 12+ \varepsilon})\) for any \(\varepsilon >0\). The author shows, among other things, that the exponential function \(e^{\lambda x}\) (with some constraints on the complex number \(\lambda)\) and certain polynomials of degrees 4 and 5 fall into this class. The results constitute numerical improvements over previous investigation by \textit{M. Mikolás} [Acta Univ. Szeged., Acta Sci. Math. 13, 93-117 (1949; Zbl 0035.31402)]. The detailed results are too complicated to present here. The improvements are based on a more delicate approximation to \(\sum_{n\leq u} n^{-3}\) by \(\zeta (3) -\frac 12 (u+1)^{-2}\).