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European Journal of Mathematics

, Volume 3, Issue 2, pp 363–378 | Cite as

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

  • Matthias KunikEmail author
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

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

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.IANUniversität MagdeburgMagdeburgGermany

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