Statistics of the multiplicative group

Abstract:

The multiplicative group MnMn is the group of units in the ring Z/nZZ/nZ, that is, the group of reduced residue classes modulo nn under multiplication. It is some abelian group of order ϕ(n)ϕ(n), and many questions about its structure can be phrased as interesting number theory questions. For example, every finite abelian group is uniquely isomorphic to a direct sum of cyclic groups Cd1⊕⋯⊕CdℓCd1⊕⋯⊕Cdℓ where each djdj divides dj+1dj+1 (these djdj are the "invariant factors"). 

The largest invariant factor dℓdℓ is exactly the Carmichael lambda function value λ(n)λ(n), while the number of invariant factors is essentially ω(n)ω(n), the number of distinct prime factors of nn; both of these quantities have been thoroughly studied by analytic number theorists, and in particular the precise limiting distribution of ω(n)ω(n) is known (the "Erdös–Kac theorem").

In this seminar talk, I describe various other statistics of the multiplicative groups MnMn whose distribution can be analyzed using the tools of analytic number theory. Lee Troupe and I have counted the number of subgroups of MnMn and established an analogue of the Erdös–Kac theorem for that number. Jenna Downey and I are obtaining, for any fixed pp-group HH, an asymptotic formula for the counting function of those nn for which HH is the Sylow pp-subgroup of MnMn (a generalization of the classical counting function of those nn for which p∤ϕ(n)p∤ϕ(n)). Finally, Ben Chang and I are obtaining, for any fixed integer dd, an asymptotic formula for the counting function of those nn for which the least invariant factor of MnMn equals dd.