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Discrepancy of Farey sequences. (Discrépance des suites de Farey.) (French) Zbl 0981.11026

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 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 \(10^{-100})\).

MSC:

11K38 Irregularities of distribution, discrepancy
11K31 Special sequences

Citations:

Zbl 0248.10013
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References:

[1] Cohen, H., Dress, F. et El Marraki, M., Ezplicit estimates for summatory functions linked to the Môbius m-function. Preprint A2X n° 96-7 (1996), soumis à Math. Comp.
[2] Dress, F., Fonction sommatoire de la fonction de Möbius, 1. Majorations expérimentales. Expérimental Mathematics2 (1993), 89-98. · Zbl 0817.11061
[3] Dress, F. et El Marraki, M., Fonction sommatoire de la fonction de Möbius, 2. Majorations asymptotiques élémentaires. Expérimental Mathematics2 (1993), 99-112. · Zbl 0817.11062
[4] El Marraki, M., Fonction sommatoire de la fonction de Möbius, 3. Majorations asymptotiques effectives. Journal de Théorie des Nombres de Bordeaux7 (1995), 407-433. · Zbl 0869.11075
[5] Franel, J., Les suites de Farey et le problème des nombres premiers. Gôttingen Nachrichten (1924), 198-201. · JFM 50.0119.01
[6] Niederreiter, H., The distribution of Farey points. Math. Ann.201 (1973), 341-345. · Zbl 0248.10013
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