2025/03/11 by Maxim Aleksandrovich Korolev, Korolev, Maxim A.
Mathematics · #Analytic Number Theory Research #Mathematical functions and polynomials #Advanced Mathematical Identities
paper · pdf · doi:10.48550/arxiv.2503.08176
This version corrects minor inaccuracies and missprints. One drawing is changed. We continue to study some arithmetical properties of Farey sequences by the method introduced by F.Boca, C.Cobeli and A.Zaharescu (2001). Let ΦQ be the classical Farey sequence of order Q. Having the fixed integers D\geqslant 2 and 0\leqslant c\leqslant D-1, we colour to the red the fractions in ΦQ with denominators ≡ c \pmod D. Consider the gaps in ΦQ with coloured endpoints, that do not contain the fractions a/q with q≡ c \pmod D inside. The question is to find the limit proportions ν(r;D,c) (as Q→ +∞) of such gaps with precisely r fractions inside in the whole set of the gaps under considering (r = 0,1,2,3,…). In fact, the expression for this proportion can be derived from the general result obtained by C.Cobeli, M.Vâjâitu and A.Zaharescu (2014). However, such formula expresses ν(r;D,c) in the terms of areas of some polygons related to a special geometrical transform. In the present paper, we obtain explicit formulas for ν(r;D,c) for the cases 3 and c=1,2. Thus this paper cover the case D=3.