The Ramanujan Journal

, Volume 16, Issue 2, pp 131–161 | Cite as

On the index of fractions with square-free denominators in arithmetic progressions

  • Emre Alkan
  • Andrew H. Ledoan
  • Marian Vâjâitu
  • Alexandru Zaharescu
Article
  • 62 Downloads

Abstract

We prove asymptotic formulas for the first and second moments of the index of fractions with square-free denominators of order Q streaming in a given arithmetic progression as Q→∞.

Keywords

Index of Farey fractions Square-free integers Riemann zeta-function and Dirichlet L-functions Arithmetic progressions 

Mathematics Subject Classification (2000)

11B57 11M06 11N25 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Boca, F.P., Cobeli, C., Zaharescu, A.: On the distribution of the Farey sequence with odd denominators. Mich. Math. J. 51(3), 557–574 (2003) MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Boca, F.P., Gologan, R.N., Zaharescu, A.: The average length of a trajectory in a certain billiard in a flat two-torus. N.Y.  J. Math. 9, 303–330 (2003) MATHMathSciNetGoogle Scholar
  3. 3.
    Boca, F.P., Gologan, R.N., Zaharescu, A.: The statistics of the trajectory of a certain billiard in a flat two-torus. Commun. Math. Phys. 240(1–2), 53–73 (2003) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Davenport, H.: Multiplicative Number Theory, 3rd edn. Graduate Studies in Mathematics, vol. 74 (Montgomery, H.L. (ed.)). Springer, New York (2000) MATHGoogle Scholar
  5. 5.
    Elliot, P.D.T.A.: Probabilistic Number Theory: I. Mean-Value Theorems. Springer, New York (1979) Google Scholar
  6. 6.
    Hall, R.R., Shiu, P.: The index of a Farey sequence. Mich. Math. J. 51(2), 209–223 (2003) MATHMathSciNetGoogle Scholar
  7. 7.
    Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers, 5th edn. Clarendon Press, Oxford (1979) MATHGoogle Scholar
  8. 8.
    Haynes, A.: A note on Farey fractions with odd denominators. J. Number Theory 98(1), 89–104 (2003) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Haynes, A.: The distribution of special subsets of the Farey sequence. J. Number Theory 107(1), 95–104 (2004) MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Ivić, A.: The Riemann Zeta-Function. Wiley-Interscience, New York (1985) MATHGoogle Scholar
  11. 11.
    Montgomery, H.L.: Topics in Multiplicative Number Theory. Lecture Notes in Mathematics, vol. 227. Springer, New York (1971) MATHGoogle Scholar
  12. 12.
    Montgomery, H.L., Vaughan, R.C.: Multiplicative Number Theory: I. Classical Theory. Cambridge Studies in Advanced Mathematics, vol. 97. Cambridge University Press, Cambridge (2007) MATHGoogle Scholar
  13. 13.
    Titchmarsh, E.C.: The Theory of Functions, 2nd edn. Oxford University Press, London (1939) MATHGoogle Scholar
  14. 14.
    Titchmarsh, E.C.: The Theory of the Riemann Zeta-Function, 2nd edn. (Heath-Brown, D.R. (ed.)). Oxford University Press, New York (1986) MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  • Emre Alkan
    • 1
  • Andrew H. Ledoan
    • 1
  • Marian Vâjâitu
    • 2
  • Alexandru Zaharescu
    • 1
  1. 1.Department of MathematicsUniversity of Illinois at Urbana-ChampaignUrbanaUSA
  2. 2.“Simion Stoilow” Institute of MathematicsRomanian AcademyBucharestRomania

Personalised recommendations