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The least primary factor of the multiplicative group

Greg Martin, Chau Nguyen

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.

The least primary factor of the multiplicative group

Abstract

Let denote the least primary factor in the primary decomposition of the multiplicative group . We give an asymptotic formula, with order of magnitude , for the counting function of those integers for which . We also give an asymptotic formula, for any prime power , for the counting function of those integers for which . 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.
Paper Structure (8 sections, 15 theorems, 60 equations)

This paper contains 8 sections, 15 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Counting integers $n$ with least primary factor not equal to $2$
  3. Counting integers $n$ with least primary factor equal to $q$
  4. Plan for evaluating the leading constants
  5. Calculating the leading constant $C_3$
  6. Evaluating ${\sum_{b \in {\mathcal{B}}}\chi(b)}$
  7. Calculating the leading constant for $q=4$
  8. Calculating the leading constant for $q=5$

Key Result

Theorem 1.1

The number of $n\le x$ for which $2$ is not the least primary factor of $M_n$ is where

Theorems & Definitions (46)

  • Theorem 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Lemma 2.3
  • proof
  • ...and 36 more