与广义Dedekind函数有关的误差项估计

Mean value estimates of an error term related to generalized Dedekind totient function.

设ψ(n)是Dedekind函数.r为正整数,则有∑n≤xnψ(n)r=αx+E(x,r),其中α是与r有关的常数,而E(x,r)是误差项.利用经典的复积分理论及解析的方法研究了E(x,r)的算术均值和积分均值,得到了一个较为精确的估计式.

Let ψ(n) is the Dedekind totient function, and r is a positive integer. It is known that∑n≤xnψ(n)r=αx+E(x,r),where α is a constant related to r and E(x,r) is the error term. The main purpose was using the classical complex integral theory and the analytic method to study the arithmetic and integral mean value of E(x,r), and gave a more precise asymptotic formula.