# The smallest invariant factor of the multiplicative group

## TL;DR

This paper provides asymptotic formulas for counting integers based on the smallest invariant factor of their multiplicative group, revealing distribution patterns with explicit error bounds.

## Contribution

It introduces new asymptotic formulas for the distribution of the smallest invariant factor in multiplicative groups, including explicit dependence in the Selberg-Delange method.

## Key findings

- Asymptotic count of integers with smallest invariant factor not equal to 2.
- Asymptotic count for integers with smallest invariant factor equal to any even q ≥ 4.
- Uniform asymptotic formula for integers with prime factors in fixed residue classes.

## Abstract

Let $\lambda_1(n)$ denote the least invariant factor in the invariant factor decomposition of the multiplicative group $M_n = (\mathbb Z/n\mathbb Z)^\times$. We give an asymptotic formula, with order of magnitude $x/\sqrt{\log x}$, for the counting function of those integers $n$ for which $\lambda_1(n)\ne2$. We also give an asymptotic formula, for any even $q\ge4$, for the counting function of those integers $n$ for which $\lambda_1(n)=q$. These results require a version of the Selberg-Delange method whose dependence on certain parameters is made explicit, which we provide in an appendix. As an application, we give an asymptotic formula for the counting function of those integers $n$ all of whose prime factors lie in an arbitrary fixed set of reduced residue classes, with implicit constants uniform over all moduli and sets of residue classes.