The Ramanujan Journal

, Volume 47, Issue 3, pp 547–564 | Cite as

Subgroups of cyclic groups and values of the Riemann zeta function

  • Mahannah El-Farrah
  • Dominic LanphierEmail author


Let k be a positive integer, x a large real number, and let \(C_n\) be the cyclic group of order n. For \(k\le n\le x\) we determine the mean average order of the subgroups of \(C_n\) generated by k distinct elements and we give asymptotic results of related averaging functions of the orders of subgroups of cyclic groups. The average order is expressed in terms of Jordan’s totient functions and Stirling numbers of the second kind. We have the following consequence. Let k and x be as above. For \(k\le n\le x\), the mean average proportion of \(C_n\) generated by k distinct elements approaches \(\zeta (k+2)/\zeta (k+1)\) as x grows, where \(\zeta (s)\) is the Riemann zeta function.


Cyclic groups Riemann zeta function Jordan’s totient functions 

Mathematics Subject Classification

Primary 20P05 11A25 Secondary 11B73 


  1. 1.
    Adhikari, S.D., Sankaranarayanan, A.: On an error term related to the Jordan totient function \(J_k(n)\). J. Number Theory 34, 178–188 (1990)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Amiri, H., Amiri, S.M.J., Isaacs, I.M.: Sums of element orders in finite groups. Commun. Algebra 37(9), 2978–2980 (2009)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Amiri, H., Amiri, S.M.J.: Sums of element orders on finite groups of the same order. J. Algebra Appl. 10(2), 187–190 (2011)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Amiri, H., Amiri, S.M.J.: Sums of element orders of maximal subgroups of the symmetric group. Commun. Algebra 40(2), 770–778 (2012)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Babai, L.: The probability of generating the symmetric group. J. Combin. Theory Ser. A 52(1), 148–153 (1989)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Beardon, A.F.: Sums of powers of integers. Am. Math. Mon. 103, 201–213 (1996)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Dixon, J.D.: The probability of generating the symmetric group. Math. Z. 110, 199–205 (1969)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Dixon, J.D.: Asymptotics of generating the symmetric and alternating groups. Electron. J. Combin. 12 (2005), Research Paper 56Google Scholar
  9. 9.
    Edwards, A.W.F.: Sums of powers of integers: a little of the history. Math. Gaz. 66, 22–29 (1982)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Erdős, P., Turán, P.: On some problems of a statistical group theory IV. Acta Math. Acad. Scient. Hung. 19, 413–435 (1968)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Gallian, J.: Contemporary Abstract Algebra, 6th edn. Houghton Mifflin Company, Boston (2006)zbMATHGoogle Scholar
  12. 12.
    Gathen, J., Knopfmacher, A., Luca, F., Lucht, L.G., Shparlinski, I.E.: Average order in cyclic groups. J. de Théorie des Nombres de Bordeaux 16, 107–123 (2004)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Goh, W.M., Schmutz, E.: The expected order of a random permutation. Bull. Lond. Math. Soc. 23(1), 34–42 (1991)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Gould, H.W., Shonhiwa, T.: Functions of GCD’s and LCM’s. Indian J. Math. 39(1), 11–35 (1997)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Gould, H.W., Shonhiwa, T.: A generalization of Cesàro’s function and other results. Indian J. Math. 39(2), 183–194 (1997)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers, 6th edn. Oxford University Press, Oxford (2008)zbMATHGoogle Scholar
  17. 17.
    Harrington, J., Jones, L., Lamarche, A.: Characterizing finite groups using the sum of the orders of the elements. Int. J. Comb. vol. (2014), Article 835125Google Scholar
  18. 18.
    Hu, Y., Pomerance, C.: The average order of elements in the multiplicative group of a finite field. Involv. J. Math. 5(2), 229–236 (2012)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Luca, F.: Some mean values related to average multiplicative orders of elements in finite fields. Ramanujan J. 9, 33–44 (2005)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Marefat, Y., Iranmanesh, A., Tehranian, A.: On the sum of element orders of finite simple groups. J. Algebra Appl. 10(2) (2013)Google Scholar
  21. 21.
    McCarthy, J.P.: Introduction to Arithmetical Functions. Springer, New York (1986)CrossRefGoogle Scholar
  22. 22.
    Rosen, K.H.: Discrete Mathematics and Its Applications, 7th edn. McGraw-Hill, New York (2007)Google Scholar
  23. 23.
    Schmutz, E.: Proof of a conjecture of Erdős and Turán. J. Number Theory 31(3), 260–271 (1989)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Spencer, J.: (with L. Florescu), Asymptopia (Student Mathematical Library), vol. 71. American Mathematical Society, Providence (2014)Google Scholar
  25. 25.
    Stanley, R.P.: Enumerative Combinatorics. Vol. 1, Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1997)Google Scholar
  26. 26.
    Walfisz, A.: Weylsche Exponentialsummen in der neueren Zahlentheorie, Mathematische Forschungsberichte 15 VEB Deutscher Verlag der Wissenschaften, Berlin (1963)Google Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsWestern Kentucky UniversityBowling GreenUSA

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