Discrepancy of Farey sequences
The author proves the very remarkable result, that the absolute discrepancy of the Farey sequence of order \(n\), consisting of all rational numbers in the unit interval of denominator smaller or equal to \(n\), is exactly \(1/n\). The right order of growth was already obtained in \textit{H. Niederreiter} [Math. Ann. 201, 341-345 (1973; Zbl 0248.10013)]. An analogous result for the square mean discrepancy is equivalent to the Riemann hypothesis by an old result of Franel. The proof given here is divided into two difficult parts \((n< 10^{110}\) and \(N> 10^{400})\), studies ``major and minor arcs'' and uses upper bounds of an integral related to a summatory function of the Möbius function. One has to estimate carefully several special cases, in particular in connection with rationals of small denominators. Useful is also a concept of truncated convergence, neglecting ``small relative errors'' (of order smaller than \(10^{-100})\).
scientific article; zbMATH DE number 1426376