Skip to main content
SpringerLink
Log in

A congruence sum and rational approximations

  • Published:
Rendiconti del Circolo Matematico di Palermo Series 2 Aims and scope Submit manuscript

Abstract

We give a reciprocity formula for a two-variables sum where the variables satisfy a linear congruence condition. We also prove that such sum is a measure of how well a rational is approximable from below and show that the reciprocity formula is a simple consequence of this fact.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

Notes

  1. Notice that if q is prime, then the average of S(a / q) reduces exactly to the Dirichlet’s divisor problem: \(\frac{1}{\varphi (q)}\sum _{\begin{array}{c} 0<a<q,\\ (a,q)=1 \end{array}}S(a/q)=\sum _{n<q}d(n)\), where d(n) is the number of divisors of n.

References

  1. Bettin, S.: On the distribution of a cotangent sum. Int. Math. Res. Not. 2015(21), 11419–11432 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bettin, S.: On the reciprocity law for Dirichlet \(L\)-functions. Trans. Am. Math. Soc. 368, 6887–6914 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  3. Conrey, J.B.: The mean-square of Dirichlet L-functions. arXiv:0708.2699 [math.NT]

  4. Conrey, J.B., Keating, J.P.: Moments of Zeta and Correlations of Divisor-Sums: II. Advances in the Theory of Numbers. In: The Series Fields Institute Communications. vol. 77, pp. 75–85

  5. Heilbronn, H.: On the average length of a class of finite continued fractions. In: Number Theory and Analysis (Papers in Honor of Edmund Landau), pp. 87–96. Plenum, New York (1969)

  6. Hickerson, D.: Continued fractions and density results for Dedekind sums. J. Reine Angew. Math. 290, 113–116 (1977)

    MathSciNet  MATH  Google Scholar 

  7. Khinchin, A.Y.: Continued Fractions. The University of Chicago Press, Chicago (1964)

    MATH  Google Scholar 

  8. Vardi, I.: Dedekind sums have a limiting distribution. Int. Math. Res. Not. 1993(1), 1–12 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  9. Young, M.P.: The reciprocity law for the twisted second moment of Dirichlet L-functions. Forum Math. 23(6), 1323–1337 (2011)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This note was mostly written while the author was a postdoctoral fellow at the Centre de recherches mathématiques in Montréal.

Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Università di Genova, via Dodecaneso 35, 16146, Genoa, Italy

    Sandro Bettin

Authors
  1. Sandro Bettin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sandro Bettin.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bettin, S. A congruence sum and rational approximations. Rend. Circ. Mat. Palermo, II. Ser 66, 477–483 (2017). https://doi.org/10.1007/s12215-016-0288-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12215-016-0288-0

Keywords

Mathematics Subject Classification

Search

Navigation