For a given positive integer ℓ we show the existence of the limiting gap distribution measure for the sets of Farey fractions a q of order Q with ℓ ∤ a, and respectively with (q, ℓ) = 1, as Q → ∞.

Abstract. In [3] and [2], Atanassov introduced the two arithmetic functionscalled the irrational factor and the restrictive factor, respectively. Alkan, Ledoan, Panaitopol, and the authors explore properties of these arithmetic functions in [1], [7], [8] and [9]. In the present paper, we generalize these functions to a larger class of elements of PSL 2 (Z), and explore some of the properties of these maps.

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Abstract. Atanassov introduced the irrational factor function and the strong restrictive factor function, which he defined asand [3]. Various properties of these functions have been investigated by Alkan, Ledoan, Panaitopol, and the authors. In the prequel, we expanded these functions to a class of elements of PSL 2 (Z), and studied some of the properties of these maps. In the present paper we generalize the previous work by introducing real moments and considering a larger class of maps. This allows us to explore new properties of these arithmetic functions.