Abstract
We derive an asymptotic formula for the number of pairs of consecutive fractions a0=q0 and a=q in the Farey sequence of order Q such that a=q; q=Q; and (Q - q0)=q lie each in prescribed subintervals of the interval [0; 1]. We deduce the leading term in the asymptotic formula for `the hyperbolic lattice point problem" for the modular group PSL(2; Z ), the number of images of a given point under the action of the group in a given circle in the hyperbolic plane.
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