关于Euler函数φ(n)的一个均值估计

ON A SUM OF EULER'S TOTIET FUNCTION

本文用解析方法得到了均值估计sum from n≥3 to n≤x 1/logφ(n)=x sum from j=1 to a-a_j/log^jx+O(x/log^(a+1)x)其中φ(n)是Euler函数,a为任意自然数,a_1=1,a_2=1-sum from p 1/plog(1-1/p),一般地 a_j=(-1)^(j-1)E^(j-1)(t)|t=0这里 E(t)=1/(t+1) multiply from p(1-1/p)(1+1/p(1-1/p)^(t-1))

Euler's totient function, φ(n), is the numbor of numbers not greater than n and prime to n. By an elementary method, one can get the following asymptotic formula sum from 3≤n≤x (1/(logφ(n))=x/(logx)(1+O(1)) In this paper, by an analytical method, we get the following asymototic expression. Theorem. Let a be an arbitrary positive integer, then sum from 3≤n≤x (1/(logφ(n)))=x sum from f=1 to a ((a_i)/(logix))+O(x/log^(a+1)x where a_1=1,a_2=1-sum from p= 1/p log(1-1/p), in genetal, a_i=(-1)^(j-1)E^(j-1)(t)|1=0 with E(1)=1/(t+1) (?)(1-1/p)(1+1/p(1-1/p)^((?)-1))