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

Abstract

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

Thirteen Pages. Keywords: Multiplicative function; Average orders; Euler phi function; Dedekind psi function, Sum of divisors function; Largest integer function

Cited by in corpus (1)

• Sum of Divisors Function And The Largest Integer Function Over The Shifted Primes