Euler φ函数、Dedekind ψ函数、因子和函数和最大函数的平均阶
Average Orders of the Euler Phi Function, The Dedekind Psi Function, The Sum of Divisors Function, And The Largest Integer Function
摘要：
设$ x\geq 1 $是一个大的数，设$ [x]=x-\{x\} $是最大的整数函数，设$ φ(n)$是欧拉函数。结果$ \sum_{n\leq x}φ([x/n])=(6/π^2)x\log x+O\left ( x(\log x)^{2/3}(\log\log x)^{1/3}\right ) $最近得到了证明。本文给出了一个简短的初等证明，并将误差项锐化为$ \sum_{n\leq x}φ([x/n])=(6/π^2)x\log x+O(x) $。此外，本文还对有限和$ \sum_{n\leq x}ψ([x/n])=(15/π^2)x\log x+O(x\log \log x) $和$ \sum_{n\leq x}σ([x/n])=(π^2/6)x\log x+O(x \log \log x) $的渐近公式的第一次证明进行了评价.
英文摘要：
Let $ x\geq 1 $ be a large number, let $ [x]=x-\{x\} $ be the largest integer function, and let $ φ(n)$ be the Euler totient function. The result $ \sum_{n\leq x}φ([x/n])=(6/π^2)x\log x+O\left ( x(\log x)^{2/3}(\log\log x)^{1/3}\right ) $ was proved very recently. This note presents a short elementary proof, and sharpen the error term to $ \sum_{n\leq x}φ([x/n])=(6/π^2)x\log x+O(x) $. In addition, the first proofs of the asymptotics formulas for the finite sums $ \sum_{n\leq x}ψ([x/n])=(15/π^2)x\log x+O(x\log \log x) $, and $ \sum_{n\leq x}σ([x/n])=(π^2/6)x\log x+O(x \log \log x) $ are also evaluated here.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
