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The h-spacing distribution between Farey points


Let I = [α, β] be a subinterval of [0, 1]. For each positive integer Q, we denote by [Fscr ]I(Q) the set of Farey fractions of order Q from I, that is

and order increasingly its elements γj = aj/qj as α [les ] γ1 < γ2 < … < γNI(Q) [les ] β. The number of elements of [Fscr ]I(Q) is

We simply let [Fscr ](Q) = [Fscr ][0,1](Q), N(Q) = N[0,1](Q).

Farey sequences have been studied for a long time, mainly because of their role in problems related to diophantine approximation. There is also a connection with the Riemann zeta function which has motivated their study. Farey sequences seem to be distributed as uniformly as possible along [0, 1]; a way to prove it is to show that

for all ε > 0, as Q → ∞. Yet this is a very strong statement, as Franel and Landau [3, 4] have shown that (1·3) is equivalent to the Riemann Hypothesis.

Our object here is to investigate the distribution of spacings between Farey points in subintervals of [0, 1]. Various results related to this problem have been obtained by [2, 3, 5–8, 10–13].

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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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