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The multiplicative group $M_n$ is the group of units in the ring $\Bbb Z/n\Bbb Z$, that is, the group of reduced residue classes modulo $n$ under multiplication. It is some abelian group of order $\phi(n)$, 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 $C_{d_1} \oplus \cdots \oplus C_{d_\ell}$ where each $d_j$ divides $d_{j+1}$ (these $d_j$ are the "invariant factors"). The largest invariant factor $d_\ell$ is exactly the Carmichael lambda function value $\lambda(n)$, while the number of invariant factors is essentially $\omega(n)$, the number of distinct prime factors of $n$; both of these quantities have been thoroughly studied by analytic number theorists, and in particular the precise limiting distribution of $\omega(n)$ is known (the "Erdös–Kac theorem").

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