On the number of abelian groups of a given order (supplement)

• Autorzy: Hong-Quan Liu
• Języki publikacji: EN

Abstrakty

EN

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]).

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 1993
• Tom: 64
• Numer: 3
• Strony: 285-296

Daty

wydano

1993

otrzymano

1992-10-27

Twórcy

autor

Hong-Quan Liu

• 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.