## Abstract

Let I = (α, β] ⊆ (0, 1] be an interval, let γ_{1}, < γ_{2} < ⋯ < γ_{NI(Q)} denote the Farey fractions of order Q from I and set S_{r}(Q) = ∑_{j=1}^{NI(Q)} (γ_{j+1} - γ_{j})(γ_{j+r+1} - γ_{j+r}) for r ≧ 0. We prove that, for all r ≧ 2, there exists a constant I_{r} such that the following asymptotic formula holds: S_{r}(Q) = S_{r}(Q, (0,1)] = 6I_{r}/π^{2} Q^{2} + O_{r} (log ^{1/(r+3)}Q/Q^{2+1/(r+3)} This implies, for all h ≧ 3, the existence of a constant D(h) such that ∑_{j=1}^{N(0,1)(Q)} (γ_{j+h}-γ_{j})^{2} = 12(2h-1) logQ/π^{2} Q^{2} + D(h)/Q^{2} + O_{h}(log^{1/(h+2)} Q/Q^{2+1/(h+2)}) providing a positive answer to a conjecture of R. R. Hall. Furthermore, we prove that, for any r ≧ 2 and any such interval I, the sums S_{r}(Q, I) and S_{r}(Q) are related by S_{r}(Q, I) = |I|S_{r}(Q) + O_{ε}(Q^{-2-1/(2r+4)+ε}). For r ∈ {0, 1} we prove, for any I with rational end points, the existence of a constant c_{I} such that S_{0}(Q,I) = |I|S_{0}(Q) + 2_{CI}/Q^{2} + O_{ε}(Q^{-2-1/10+ε}) and S_{1}(Q,I) = |I|S_{I}(Q) + C_{I}/Q^{2} + O_{ε}(Q^{-2-1/12+ε}).

Original language | English (US) |
---|---|

Pages (from-to) | 207-236 |

Number of pages | 30 |

Journal | Journal fur die Reine und Angewandte Mathematik |

Volume | 535 |

DOIs | |

State | Published - 2001 |

Externally published | Yes |

## ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics