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A scaling property of Farey fractions. Part IV: mean value formulas

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Abstract

The Farey sequence of order n consists of all reduced fractions a / b between 0 and 1 with positive denominator b less or equal to n. In a series of former papers we obtained a limit function which describes a scaling property of the Farey sequence of order n for \(n \rightarrow \infty \) in the vicinity of any fixed fraction a / b and which is independent of a / b. In this paper we derive a representation formula for a sequence of functions used in the Franel–Landau theorem to determine the number of Farey-fractions in prescribed intervals, and we establish a corresponding limit function for this sequence as well. This is combined with a former result to derive a remarkable representation formula for arithmetic means of related functions.

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  1. IAN, Universität Magdeburg, Gebäude 02, Universitätsplatz 2, 39106, Magdeburg, Germany

    Matthias Kunik

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  1. Matthias Kunik
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Correspondence to Matthias Kunik.

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Kunik, M. A scaling property of Farey fractions. Part IV: mean value formulas. European Journal of Mathematics 4, 1549–1559 (2018). https://doi.org/10.1007/s40879-017-0197-6

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  • DOI: https://doi.org/10.1007/s40879-017-0197-6

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