Subgroups of cyclic groups and values of the Riemann zeta function

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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.

Keywords

Cyclic groups Riemann zeta function Jordan’s totient functions 

Mathematics Subject Classification

Primary 20P05 11A25 Secondary 11B73 

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Authors and Affiliations

  1. 1.Department of MathematicsWestern Kentucky UniversityBowling GreenUSA

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