Farey series and the Riemann hypothesis. IV (Q5932722)

This paper improves on part II in the series [Acta Math. Hung. 78, 287-304 (1978; Zbl 0902.11034)]. Consider the error term \[ E_f(x) := \sum_{\nu =1}^{\Phi(x)}f(\rho_\nu) - \Phi(x)\int_0^1 f(t) dt, \] where \(f(t)\) is an integrable function on \([0,1]\), \(\rho_\nu\) is the \(\nu^{\text{th}}\) Farey fraction of order \(x\) denoted by \({b_\nu \over c_\nu}\), \((b_\nu , c_\nu)=1\), \(0<b_\nu \leq c_\nu \leq x\), and \(\Phi(x)\) is the number of such Farey fractions. It is well-known that this error term depends strongly on the horizontal position of the zeros of the Riemann zeta-function, but it also depends on the smoothness of \(f\). The property about this error term we would like to prove for as large a class of functions as possible is that the Riemann hypothesis is equivalent to the estimate \(E_f(x) = O(x^{{1\over 2}+\epsilon})\) for every \(\epsilon>0\). Denote this statement by (E). Let \(\Lambda_\alpha\) denote the class of Lipschitz functions of order \(\alpha\), with \(0<\alpha \leq 1\). Then in part II it was proved that for \(f\in \Lambda_\alpha\), \(E_f(x) = O(x^{2-\alpha})\), and assuming the Riemann hypothesis, \(E_f(x) = O(x^{2-{3\over 2}\alpha +\varepsilon})\). Thus one direction of the equivalence in (E) is valid when \(\alpha =1\) (which includes the class of functions with bounded derivatives) and it is the opposite direction where a zero-free region must be obtained from an estimate for \(E_f\) that requires attention. The author first presents theorems which are numerical improvements on the results in part II. For example, Theorem 1 states that (E) is valid for \(f(t) = e^{\lambda t}\), \(\lambda = \lambda_1 + i \lambda_2 \neq 0\), \(|\lambda|^2 \leq 20/C\) if \(|\lambda_1|\geq |\lambda_2|\), and \((C-5/\pi^2){\lambda_1}^2 + (C+5/{\pi}^2){\lambda_2}^2 \leq 20\) if \(|\lambda_1|\leq |\lambda_2|\) with \(C= {\zeta(3/2)\zeta(7/2)\over 3 \zeta(3)}\). There are further numerical improvements on results for some polynomials of degree 4 and 5, and for functions which are three times differentiable. The author also proves two new results. A special case of the first result is that (E) holds for \(f(t) = \log \Gamma(t+\lambda)\), \(\lambda \geq \sqrt{C/20} = 0.202018 \), where \(\Gamma\) is the gamma function and \(C\) is as in Theorem 1. The second result is that (E) holds for \(f(t) = t^k\), \(2\leq k \leq 7\).