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European Journal of Mathematics

, Volume 3, Issue 2, pp 363–378

# A scaling property of Farey fractions. Part III: representation formulas

Research Article

## 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. The sums of the inverse denominators 1 / b of the Farey fractions in prescribed intervals with rational bounds have simple main terms, whereas the deviations are determined by a sequence of polygonal functions $$f_n$$. In a former paper we obtained a limit function for $$n \rightarrow \infty$$ which describes an asymptotic scaling property of functions $$f_n$$ in the vicinity of any fixed fraction a / b and which is independent of a / b. In this paper we derive new representation formulas for $$f_n$$ and related functions which give much better remainder term estimates. We also combine these results with those from our previous papers in order to prove that the sequence of functions $$f_n$$ converges pointwise to zero.

## Keywords

Farey sequences Riemann zeta function Prime number theorem Diophantine approximation

## Mathematics Subject Classification

11B57 11M06 11N05 11K60

## References

1. 1.
Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers, 5th edn. Oxford University Press, New York (1979)
2. 2.
Kunik, M.: A scaling property of Farey fractions. Eur. J. Math. 2(2), 383–417 (2016)
3. 3.
Kunik, M.: A scaling property of Farey fractions. Part II: convergence at rational points. Eur. J. Math. 2(3), 886–896 (2016)
4. 4.
Prachar, K.: Primzahlverteilung. Grundlehren der Mathematischen Wissenschaften, vol. 91. Springer, Berlin (1978)

## Copyright information

© Springer International Publishing AG 2017

## Authors and Affiliations

• Matthias Kunik
• 1
1. 1.IANUniversität MagdeburgMagdeburgGermany

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