## 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

### Keywords:

absolute discrepancy; Farey sequence of order $$n$$

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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