Dedekind函数ψ(n)的误差项的性质

On an Error Term Related to the Dedekind Totient Function

Dedekind函数是一个比较重要的算术函数。许多作者都证明了sum from n≤x φ(n)=(ζ(2)/2ζ(4))x^2+O(xlogx)。以E(X)记上式中误差项。本文首先改进了上式对误差项E(x)的估计,其次研究了E(x)的算术均值和积分均值,最后证明了E(X)的一个Ω-结果。

Dedekind totient function ψ (n) is an important arithmatic function. It has been proved by many authors that sum from n x φ(n)=(ζ(2)/2ζ(4))x^2+O(xlogx). Denote by E(x) the error term in the above formula, In this paper, we sharpen the above estimate for E(x), and establish asymptotic formulae for sum from n≤x E(n) and integral from 1 to x E(t)dt, from which an Ω-resuit for E(x) is deduced.