Dedekind函数k次方ψ^k（n）的均值估计

ON THE MEAN VALUE ESTIMATE FOR THE kTH POWER OF THE DEDEKIND TOTIENT FUNCTOIN ψ k(n)

设ψ（ｎ）是Ｄｅｄｅｋｉｎｄ函数，给出了ｋ是自然数且ｋ≥２时的ψｋ（ｎ）的算术均值：ｎ≤ｘψｋ（ｎ）＝ｃ０ｘｋ＋１＋Ｏ（（ｘｌｏｇｘ）ｋ（ｌｏｇｌｏｇｘ）ｋ－１２），ｎ≤ｘ１ψｋ（ｎ）＝ｃ１＋ｃ２ｘｋ－１＋Ｏ１ｘｋ（ｌｏｇｘ）ｋ．

Let ψ(n) denote the Dedekind totient function,the mean value estimate of ψ k(n) is given: n≤xψ k(n)=c 0x k+1 +O((xlogx) k)(loglogx) k-12 ), n≤x1ψ k(n)=c 1+c 2x k-1 +O1x k(logx) k. where k is a integer and k≥2, c 0=1k+1 p1-1p1+1p k, c 1=p1+1(p+1) k-1p kζ(k),c 2=11-k p1+1ppp+1 k-1.