Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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)
Amiri, H., Amiri, S.M.J., Isaacs, I.M.: Sums of element orders in finite groups. Commun. Algebra 37(9), 2978–2980 (2009)
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)
Amiri, H., Amiri, S.M.J.: Sums of element orders of maximal subgroups of the symmetric group. Commun. Algebra 40(2), 770–778 (2012)
Babai, L.: The probability of generating the symmetric group. J. Combin. Theory Ser. A 52(1), 148–153 (1989)
Beardon, A.F.: Sums of powers of integers. Am. Math. Mon. 103, 201–213 (1996)
Dixon, J.D.: The probability of generating the symmetric group. Math. Z. 110, 199–205 (1969)
Dixon, J.D.: Asymptotics of generating the symmetric and alternating groups. Electron. J. Combin. 12 (2005), Research Paper 56
Edwards, A.W.F.: Sums of powers of integers: a little of the history. Math. Gaz. 66, 22–29 (1982)
Erdős, P., Turán, P.: On some problems of a statistical group theory IV. Acta Math. Acad. Scient. Hung. 19, 413–435 (1968)
Gallian, J.: Contemporary Abstract Algebra, 6th edn. Houghton Mifflin Company, Boston (2006)
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)
Goh, W.M., Schmutz, E.: The expected order of a random permutation. Bull. Lond. Math. Soc. 23(1), 34–42 (1991)
Gould, H.W., Shonhiwa, T.: Functions of GCD’s and LCM’s. Indian J. Math. 39(1), 11–35 (1997)
Gould, H.W., Shonhiwa, T.: A generalization of Cesàro’s function and other results. Indian J. Math. 39(2), 183–194 (1997)
Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers, 6th edn. Oxford University Press, Oxford (2008)
Harrington, J., Jones, L., Lamarche, A.: Characterizing finite groups using the sum of the orders of the elements. Int. J. Comb. vol. (2014), Article 835125
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)
Luca, F.: Some mean values related to average multiplicative orders of elements in finite fields. Ramanujan J. 9, 33–44 (2005)
Marefat, Y., Iranmanesh, A., Tehranian, A.: On the sum of element orders of finite simple groups. J. Algebra Appl. 10(2) (2013)
McCarthy, J.P.: Introduction to Arithmetical Functions. Springer, New York (1986)
Rosen, K.H.: Discrete Mathematics and Its Applications, 7th edn. McGraw-Hill, New York (2007)
Schmutz, E.: Proof of a conjecture of Erdős and Turán. J. Number Theory 31(3), 260–271 (1989)
Spencer, J.: (with L. Florescu), Asymptopia (Student Mathematical Library), vol. 71. American Mathematical Society, Providence (2014)
Stanley, R.P.: Enumerative Combinatorics. Vol. 1, Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1997)
Walfisz, A.: Weylsche Exponentialsummen in der neueren Zahlentheorie, Mathematische Forschungsberichte 15 VEB Deutscher Verlag der Wissenschaften, Berlin (1963)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
El-Farrah, M., Lanphier, D. Subgroups of cyclic groups and values of the Riemann zeta function. Ramanujan J 47, 547–564 (2018). https://doi.org/10.1007/s11139-018-9992-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11139-018-9992-z