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A scaling property of Farey fractions. Part IV: mean value formulas
- Source :
- European Journal of Mathematics. 4:1549-1559
- Publication Year :
- 2017
- Publisher :
- Springer Science and Business Media LLC, 2017.
-
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.
Details
- ISSN :
- 21996768 and 2199675X
- Volume :
- 4
- Database :
- OpenAIRE
- Journal :
- European Journal of Mathematics
- Accession number :
- edsair.doi...........44dec20cf017e99d1f7cdd69da2f2006