# 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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topHong-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 -

## References

top- [1] E. Fouvry and H. Iwaniec, Exponential sums with monomials, J. Number Theory 33 (1989), 311-333. Zbl0687.10028
- [2] H.-Q. Liu, On the number of abelian groups of a given order, Acta Arith. 59 (1991), 261-277. Zbl0737.11024

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