正整数的素因子个数的k次均值估计

k-th Mean Value Estimate of Number of Prime Divisors of Integer

令ω(n)表示正整数n的不同素因子的个数,考虑ω(n)的k次均值,运用Nathanson和Tur偄n的方法,证明了对x≥2和正整数k,有∑n≤xω(n)k=x(lnlnx)k+O(x(lnlnx)k-1),以及对每个δ>0和正整数k,使不等式ω(n)k-(lnlnn)k≥(lnlnx)k-12+δ成立的正整数n≤x的个数是O(x)。这两个结果是对ω(n)经典均值估计的推广。

Let ω(n) be the number of different prime divisors of positive integer n.With respect to the k-th power mean value of ω(n)and by means of the methods of Nathanson and Turán are utilized to prove: for x≥2 and positive integer k,∑n≤xω(n)k=x(lnlnx)k+O(x(lnlnx)~k-1 ),also for every δ>0 and positive integer k,the number of integers n≤x such thatω(n)k-(lnlnn)k≥(lnlnx)~k-1/2+δ is O(x).These results are the generalization of the classical results on ω(n).