# On the number of abelian groups of a given order (supplement)

Acta Arithmetica (1993)

• Volume: 64, Issue: 3, page 285-296
• ISSN: 0065-1036

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## Abstract

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1. Introduction. The aim of this paper is to supply a still better result for the problem considered in [2]. Let A(x) denote the number of distinct abelian groups (up to isomorphism) of orders not exceeding x. We shall prove Theorem 1. For any ε > 0, $A\left(x\right)=C₁x+C₂{x}^{1/2}+C₃{x}^{1/3}+O\left({x}^{50/199+\epsilon }\right)$, where C₁, C₂ and C₃ are constants given on page 261 of [2]. Note that 50/199=0.25125..., thus improving our previous exponent 40/159=0.25157... obtained in [2]. To prove Theorem 1, we shall proceed along the line of approach presented in [2]. The new tool here is an improved version of a result about enumerating certain lattice points due to E. Fouvry and H. Iwaniec (Proposition 2 of [1], which was listed as Lemma 6 in [2]).

## How to cite

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Hong-Quan Liu. "On the number of abelian groups of a given order (supplement)." Acta Arithmetica 64.3 (1993): 285-296. <http://eudml.org/doc/206551>.

@article{Hong1993,
abstract = {1. Introduction. The aim of this paper is to supply a still better result for the problem considered in [2]. Let A(x) denote the number of distinct abelian groups (up to isomorphism) of orders not exceeding x. We shall prove Theorem 1. For any ε > 0, $A(x) = C₁x + C₂x^\{1/2\} + C₃x^\{1/3\} + O(x^\{50/199+ε\})$, where C₁, C₂ and C₃ are constants given on page 261 of [2]. Note that 50/199=0.25125..., thus improving our previous exponent 40/159=0.25157... obtained in [2]. To prove Theorem 1, we shall proceed along the line of approach presented in [2]. The new tool here is an improved version of a result about enumerating certain lattice points due to E. Fouvry and H. Iwaniec (Proposition 2 of [1], which was listed as Lemma 6 in [2]).},
author = {Hong-Quan Liu},
journal = {Acta Arithmetica},
keywords = {asymptotic results; number of non-isomorphic abelian groups},
language = {eng},
number = {3},
pages = {285-296},
title = {On the number of abelian groups of a given order (supplement)},
url = {http://eudml.org/doc/206551},
volume = {64},
year = {1993},
}

TY - JOUR
AU - Hong-Quan Liu
TI - On the number of abelian groups of a given order (supplement)
JO - Acta Arithmetica
PY - 1993
VL - 64
IS - 3
SP - 285
EP - 296
AB - 1. Introduction. The aim of this paper is to supply a still better result for the problem considered in [2]. Let A(x) denote the number of distinct abelian groups (up to isomorphism) of orders not exceeding x. We shall prove Theorem 1. For any ε > 0, $A(x) = C₁x + C₂x^{1/2} + C₃x^{1/3} + O(x^{50/199+ε})$, where C₁, C₂ and C₃ are constants given on page 261 of [2]. Note that 50/199=0.25125..., thus improving our previous exponent 40/159=0.25157... obtained in [2]. To prove Theorem 1, we shall proceed along the line of approach presented in [2]. The new tool here is an improved version of a result about enumerating certain lattice points due to E. Fouvry and H. Iwaniec (Proposition 2 of [1], which was listed as Lemma 6 in [2]).
LA - eng
KW - asymptotic results; number of non-isomorphic abelian groups
UR - http://eudml.org/doc/206551
ER -

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