关于乘法分拆的计数函数  

ON THE ENUMERATION FUNCTION OF MULTIPLICATIVE PARTITIONS

作　　者：曹惠中[1] 

机构地区：[1]山东大学

出　　处：《数学杂志》1991年第2期183-187,共5页Journal of Mathematics

摘　　要：本文讨论了乘法分拆的计数函数 g(n)并对 g(n)的均值作了下界的估值。一 引言考虑集合 T(n)={(m_1,m_2,…,m_s);n=m_1m_2…m_s,m_i>1,1≤i≤s},此处不计m_1,m_2,…,m_s 的次序。我们定义 g(n)=|T(n)|并且 g(1)=1。例如 g(24)=7,因为24=3·8=3·4·2=3·2·2·2=6·4=6·2·2=12·2.在1983年,John F.Hughes 和 J.O.Shallit 证明了 g(n)≤2n 2^(1/2)Let g(n),n>1,be the number of ways to write n as the product of int-egers≥2,where we identify factorizations that differ only In the order of thefactors.For convenience,take g(1)=1.In this paper,we prove that sum from n≤x g(n)≥(1/384-∈)x log^3x,when x is sufficiently large,by means of an elementaryinequality,i.e.sum from n≤x g(n)≥sum from x^(1/2)<d<x sum from m≤x/d g(m).Finally,we point out if we makeuse of a result of Riemann Zeta function,i.e.D_k(x)=sum from n≤(?) τ_k(n)=xp_k(log x)+O(x^(1(1/k)) log^(k-2)x),k=2,3,…,where (?)~k(s)=sum from n 1 to ∞ (τ_k(n))/(n^(?)) and p_k is a polynomialof degree k-1,then the lower bound can be improved.

关 键 词：乘法分析 计数函数 均值 下界估计 

分 类 号：O156[理学—数学]