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

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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→∞.

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References

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  4. Davenport, H.: Multiplicative Number Theory, 3rd edn. Graduate Studies in Mathematics, vol. 74 (Montgomery, H.L. (ed.)). Springer, New York (2000)

    MATH  Google Scholar 

  5. Elliot, P.D.T.A.: Probabilistic Number Theory: I. Mean-Value Theorems. Springer, New York (1979)

    Google Scholar 

  6. Hall, R.R., Shiu, P.: The index of a Farey sequence. Mich. Math. J. 51(2), 209–223 (2003)

    MATH  MathSciNet  Google Scholar 

  7. Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers, 5th edn. Clarendon Press, Oxford (1979)

    MATH  Google Scholar 

  8. Haynes, A.: A note on Farey fractions with odd denominators. J. Number Theory 98(1), 89–104 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  9. Haynes, A.: The distribution of special subsets of the Farey sequence. J. Number Theory 107(1), 95–104 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  10. Ivić, A.: The Riemann Zeta-Function. Wiley-Interscience, New York (1985)

    MATH  Google Scholar 

  11. Montgomery, H.L.: Topics in Multiplicative Number Theory. Lecture Notes in Mathematics, vol. 227. Springer, New York (1971)

    MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  13. Titchmarsh, E.C.: The Theory of Functions, 2nd edn. Oxford University Press, London (1939)

    MATH  Google Scholar 

  14. Titchmarsh, E.C.: The Theory of the Riemann Zeta-Function, 2nd edn. (Heath-Brown, D.R. (ed.)). Oxford University Press, New York (1986)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Illinois at Urbana-Champaign, 273 Altgeld Hall, MC-382, 1409 W. Green Street, Urbana, IL, 61801-2975, USA

    Emre Alkan, Andrew H. Ledoan & Alexandru Zaharescu

  2. “Simion Stoilow” Institute of Mathematics, Romanian Academy, P.O. Box 1-764, Bucharest, 014700, Romania

    Marian Vâjâitu

Authors
  1. Emre Alkan
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  2. Andrew H. Ledoan
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  3. Marian Vâjâitu
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  4. Alexandru Zaharescu
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Corresponding author

Correspondence to Alexandru Zaharescu.

Additional information

A. Zaharescu was supported by NSF grant number DMS-0456615.

This research was also partially supported by the CERES Program 4-147/2004 of the Romanian Ministry of Education and Research.

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Alkan, E., Ledoan, A.H., Vâjâitu, M. et al. On the index of fractions with square-free denominators in arithmetic progressions. Ramanujan J 16, 131–161 (2008). https://doi.org/10.1007/s11139-007-9103-z

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  • DOI: https://doi.org/10.1007/s11139-007-9103-z

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