数论函数值分布问题中的Erds强度(Ⅰ)

ON THE ERDS'S PROPERTY OF THE DISTRIBUTION OF THE VALUES OF ARITHMETICAL FUNCTIONS (I)

本文考虑与素因子有关的加性函数ω(n)及Ω(n),证实了对于给正整数k,存在自然数n,使得在该点函数值超过左右两邻连续k个函数值的连乘积。

For the arithmetical function ω(n)=1 and Ω(n)=α_1+…+α_~ (for n=p_1^(α1)…p_k^(αk)), in this paper, we have following result: For any given positive integer n_0 and k, then there exists a natural number n, Such that when n>n_0 we have g(n)>multiply from v=1 to k g(n+v)·g(n-v), (*) here g(n)=ω(n) or Ω(n), the so-called Erdos's property of arithmetical function. More precisely, the number of integer n satisfying (*) is infinite.