Abelian theorems, Farey series and the Riemann hypothesis (Q1770965)

The present paper is a continuation of the previous three papers [Acta Math. Hung. 78, No. 4, 287--304 (1998; Zbl 0902.11034); Acta Arith. 75, No. 4, 351--374 (1996; Zbl 0860.11047); Ramanujan J. 1, No. 4, 363--378 (1997; Zbl 0908.11042)]. Here the author shows that the hypothesis \(RH (\alpha)\), that the Riemann zeta function has no zeros for \(\operatorname{Re} s\geq \alpha\), is equivalent to certain error term estimates for Farey series. For instance, it is shown that \(RH( \frac 12)\) (the Riemann Hypothesis) is equivalent to the estimate \[ \sum\limits_\nu \Big ( p^k_\nu - \frac 1{k+1}\Big ) = \mathcal O (x^{\frac 12 + \varepsilon}) \] for \(k=2, 3, \dots, 7\) where the summation is over all Farey points \(p_\nu\) of order \([x]\). This paper contains several other results of this kind. The main ingredient of the proofs is a new abelian theorem.