On local frequencies related to farey fractions

Let P1 and P2 be two sets of prime numbers and let ω(m, Pi) = #{p: p | m, p ∈ Pi} (i = 1, 2) be two related additive functions of m. For an irreducible positive fraction m/n, define h(m/n) = ω(m, P1) + ω(n, P2). In this paper the local frequencies vx{h(m/n) = s} = #{m/n ∈ F x: h(m/n) = s}/#Fx are considered, where Fx denotes the classical Farey series. Using the mean-value theorem for multiplicative functions of rational argument, a local limit theorem for v x{h(m/n) = s} is proved. © 2000 Kluwer Academic/Plenum Publishers.