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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- Hong-Quan Liu, Divisor problems of 4 and 3 dimensions
- Hong-Quan Liu, On some divisor problems
- Wenguang Zhai, Xiaodong Cao, On the average number of unitary factors of finite abelian groups
- Hong-Quan Liu, On a fundamental result in van der Corput's method of estimating exponential sums
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