  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.
