The distribution of the number of subgroups of the multiplicative group

2017/09/30 by Martin, Greg, Troupe, Lee
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1710.00124

Abstract

Let I(n) denote the number of isomorphism classes of subgroups of (\Bbb Z/n\Bbb Z)^×, and let G(n) denote the number of subgroups of (\Bbb Z/n\Bbb Z)^× counted as sets (not up to isomorphism). We prove that both log G(n) and log I(n) satisfy Erdös-Kac laws, in that suitable normalizations of them are normally distributed in the limit. Of note is that log G(n) is not an additive function but is closely related to the sum of squares of additive functions. We also establish the orders of magnitude of the maximal orders of log G(n) and log I(n).

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