Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the error term concerning the number of cyclic subgroups of Z_l \times Z_m \times Z_n with lmn\leqslant x

Jing Ma, Jiaming Li, Jia Zhang

Abstract

Let Zn denote the additive group of residue classes modulo n. Let c(l,m,n) denote the number of cyclic subgroups of Zl *Zm *Zn. For any x > 1, we consider the asymptotic behavior of D3c(x):= \sum_{lmn\leq x} c(l,m,n), obtain an asymptotic formula by complex method, and get an upper bound for the integral mean-square of the error term in that asymptotic formula.

On the error term concerning the number of cyclic subgroups of Z_l \times Z_m \times Z_n with lmn\leqslant x

Abstract

Let Zn denote the additive group of residue classes modulo n. Let c(l,m,n) denote the number of cyclic subgroups of Zl *Zm *Zn. For any x > 1, we consider the asymptotic behavior of D3c(x):= \sum_{lmn\leq x} c(l,m,n), obtain an asymptotic formula by complex method, and get an upper bound for the integral mean-square of the error term in that asymptotic formula.

Paper Structure

This paper contains 11 sections, 15 theorems, 86 equations.

Table of Contents

  1. Inteoduction
  2. Preliminary lemmas
  3. Proof of Therorem 1.1
  4. Applying Perron's Formula
  5. Evaluation of $I(x,T)$
  6. Evaluation of $J(x,T)$
  7. Evaluation of $H_j(x,T)$ for $j=1,2,3$
  8. Completion of the proof of Theorem \ref{['Theorem1']}
  9. Proof of Therorem 1.2
  10. Fourier transform
  11. Completion of the proof of Theorem \ref{['Theorem2']}

Key Result

Theorem 1.1

With the notation in D_rc, $D_{3c}(x)$ represent the total number of cyclic subgroups of finite abelian groups with order not exceed $x$ and rank not exceed $3$, we have where $P_9(u)$ is a polynomial in $u$ of degree $9$ with computable coefficients.

Theorems & Definitions (15)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.2: Perron's formula
  • Lemma 2.3: Perron's formula
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Lemma 2.8
  • ...and 5 more