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, , 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]).
@article{bwmeta1.element.bwnjournal-article-aav64i3p285bwm,
author = {Hong-Quan Liu},
title = {On the number of abelian groups of a given order (supplement)},
journal = {Acta Arithmetica},
volume = {64},
year = {1993},
pages = {285-296},
zbl = {0790.11074},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-aav64i3p285bwm}
}
Hong-Quan Liu. On the number of abelian groups of a given order (supplement). Acta Arithmetica, Tome 64 (1993) pp. 285-296. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-aav64i3p285bwm/
[000] [1] E. Fouvry and H. Iwaniec, Exponential sums with monomials, J. Number Theory 33 (1989), 311-333. | Zbl 0687.10028
[001] [2] H.-Q. Liu, On the number of abelian groups of a given order, Acta Arith. 59 (1991), 261-277. | Zbl 0737.11024