Q_k(x)与F_k(x)的渐近公式

The Asymptotic Formulae for Q_k(x) and F_k(x)

设ｎ，ｋ为自然数，Ｇ（ｎ）阶群中的同构类数，Ｆｋ（ｘ）与Ｑｋ（ｘ）分别表示不超过ｘ的自然数中使Ｇ（ｎ）＝ｋ的自然数、无平方因子自然数的个数．本文的目的是用Ｂｒｕｎ筛法证明Ｑｋ（ｘ）的条件渐近公式并对Ｆｋ（ｘ）的渐近性质做出了推测．

For natural number n and k, let G(n) denote the number of isomorphism classes of groups of order n, Fk(x) the number of n not exceeding x and G(n) = k, and Qk(x) the number of n hot exceeding x, square-free and G(n) = k. In this paper a conditional asymptotic formula is proved for Qk(x) and a conjecture is made for Fk(x) by Brun's sieve method.