Let I Y v0Y 1 be an interval, let g 1`g2`Á Á Á`g N I Q denote the Farey fractions of order Q from I and set S r Q N I Q j1 Brought to you by | provisional account Unauthenticated Download Date | 5/28/15 12:26 PM due to M. N. Huxley, was being presented. It was as well mentioned that the referee of Boca, Cobeli and Zaharescu, Conjecture on Farey points 208 Brought to you by | provisional account Unauthenticated Download Date | 5/28/15 12:26 PM Brought to you by | provisional account Unauthenticated Download Date | 5/28/15 12:26 PM Brought to you by | provisional account Unauthenticated Download Date | 5/28/15 12:26 PM X 16 Boca, Cobeli and Zaharescu, Conjecture on Farey points 211 Brought to you by | provisional account Unauthenticated Download Date | 5/28/15 12:26 PM
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 isand order increasingly its elements γj
= aj/qj as
α [les ] γ1 < γ2 < … <
γNI(Q) [les ] β.
The number of elements of [Fscr ]I(Q) isWe 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 thatfor 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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