The least primary factor of the multiplicative group

Abstract

Let $S(n)$ denote the least primary factor in the primary decomposition of the multiplicative group $M_n = (\Bbb Z/n\Bbb Z)^\times$. We give an asymptotic formula, with order of magnitude $x/(\log x)^{1/2}$, for the counting function of those integers $n$ for which $S(n) \ne 2$. We also give an asymptotic formula, for any prime power $q$, for the counting function of those integers $n$ for which $S(n) = q$. This group-theoretic problem can be reduced to problems of counting integers with restrictions on their prime factors, allowing it to be addressed by classical techniques of analytic number theory.