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MERSENNE PRIMES, IRRATIONALITY AND COUNTING SUBGROUPS

Published online by Cambridge University Press:  01 May 1997

MARCUS P. F. DU SAUTOY
MARCUS P. F. DU SAUTOY
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, 16 Mill Lane, Cambridge CB2 1SB
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Abstract

1. Introduction

Let G be a group, and write an(G) for the number of subgroups of index n in G. If G is a profinite group, we count only open subgroups. When G is finitely generated (either abstractly or topologically), an(G) is finite for all n∈N. In this case we can encode the arithmetic function n[map ]an(G) in the following zeta function:

formula here

This function was introduced by Grunewald, Segal and Smith [9] and subsequently studied in [1–6, 7, 11].

Type
Research Article
Information
Bulletin of the London Mathematical Society , Volume 29 , Issue 3 , May 1997 , pp. 285 - 294
Copyright
© The London Mathematical Society 1997

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