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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]).
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]).
Słowa kluczowe
Czasopismo
Rocznik
Tom
Numer
Strony
285-296
Opis fizyczny
Daty
wydano
1993
otrzymano
1992-10-27
Twórcy
autor
- 206-10, Bao Guo St., Harbin, 150066, P.R. China
Bibliografia
- [1] E. Fouvry and H. Iwaniec, Exponential sums with monomials, J. Number Theory 33 (1989), 311-333.
- [2] H.-Q. Liu, On the number of abelian groups of a given order, Acta Arith. 59 (1991), 261-277.
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-aav64i3p285bwm