Geodesic
On the number of abelian groups of a given order (supplement)
Acta Arithmetica, Tome 64 (1993) no. 3, pp. 285-296.

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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, $A(x) = C₁x + C₂x^{1/2} + C₃x^{1/3} + O(x^{50/199+ε})$, 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]).
DOI : 10.4064/aa-64-3-285-296

Hong-Quan Liu 1

1 
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Hong-Quan Liu. On the number of abelian groups of a given order (supplement). Acta Arithmetica, Tome 64 (1993) no. 3, pp. 285-296. doi : 10.4064/aa-64-3-285-296. http://geodesic.mathdoc.fr/articles/10.4064/aa-64-3-285-296/

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