Average Orders of the Euler Phi Function, The Dedekind Psi Function, The Sum of Divisors Function, And The Largest Integer Function

Let $x\geq 1$ be a large number, let $[x]=x-\{x\}$ be the largest integer function, and let $\varphi(n)$ be the Euler totient function. The result $\sum_{n\leq x}\varphi([x/n])=(6/\pi^2)x\log x+O\left ( x(\log x)^{2/3}(\log\log x)^{1/3}\right )$ was proved very recently. This note presents a short elementary proof, and sharpen the error term to $\sum_{n\leq x}\varphi([x/n])=(6/\pi^2)x\log x+O(x)$. In addition, the first proofs of the asymptotics formulas for the finite sums $\sum_{n\leq x}\psi([x/n])=(15/\pi^2)x\log x+O(x\log \log x)$, and $\sum_{n\leq x}\sigma([x/n])=(\pi^2/6)x\log x+O(x \log \log x)$ are also evaluated here.