Advertisement

The Ramanujan Journal

, Volume 35, Issue 1, pp 149–156 | Cite as

Averages of Ramanujan sums: note on two papers by E. Alkan

  • László Tóth
Article

Abstract

We give a simple proof and a multivariable generalization of an identity due to E. Alkan concerning a weighted average of the Ramanujan sums. We deduce identities for other weighted averages of the Ramanujan sums with weights concerning logarithms, values of arithmetic functions for gcd’s, the Gamma function, the Bernoulli polynomials, and binomial coefficients.

Keywords

Ramanujan’s sum Jordan’s function Bernoulli numbers and polynomials Gamma function 

Mathematics Subject Classification

11A25 11B68 33B15 

References

  1. 1.
    Alkan, E.: On the mean square average of special values of \(L\)-functions. J. Number Theory 131, 1470–1485 (2011)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Alkan, E.: Distribution of averages of Ramanujan sums. Ramanujan J. 29, 385–408 (2012)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Alkan, E.: Ramanujan sums and the Burgess zeta function. Int. J. Number Theory 8, 2069–2092 (2012)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Artin, E.: The Gamma Function. (translated by M. Butler), Athena Series: Selected Topics in Mathematics, Holt, Rinehart and Winston (1964)Google Scholar
  5. 5.
    Briggs, W.E., Bergman, G.M.: Problem 5091. Am. Math. Mon. 71, 334–335 (1964)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Cohen, H.: Number Theory, Vol. II. Analytic and Modern Tools. Graduate Texts in Mathematics. Springer, Berlin (2007)Google Scholar
  7. 7.
    Comtet, L.: Advanced Combinatorics. Reidel Publishing Co, The Art of Finite and Infinite Expansions. D (1974)Google Scholar
  8. 8.
    Hardy, G.H., Wright, E.M.: An introduction to the theory of numbers. In: Heath-Brown, D.R., Silverman, J.H., 6th edn. Oxford University Press (2008)Google Scholar
  9. 9.
    Liskovets, V.A.: A multivariate arithmetic function of combinatorial and topological significance. Integers 10, 155–177 (2010)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Singh, J.: Defining power sums of \(n\) and \(\varphi (n)\) integers. Int. J. Number Theory 5, 41–53 (2009)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Tóth, L.: Some remarks on Ramanujan sums and cyclotomic polynomials. Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 53(101), 277–292 (2010)MathSciNetMATHGoogle Scholar
  12. 12.
    Tóth, L.: Some remarks on a paper of V. A. Liskovets. Integers 12, 97–111 (2012)MATHGoogle Scholar
  13. 13.
    Tóth, L., Haukkanen, P.: The discrete Fourier transform of \(r\)-even functions. Acta Univ. Sapientiae Math. 3, 5–25 (2011)MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Institute of MathematicsUniversität für BodenkulturViennaAustria
  2. 2.Department of MathematicsUniversity of PécsPecsHungary

Personalised recommendations