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

TL;DR

This paper provides elementary proofs and sharper error estimates for the asymptotic behavior of sums involving the Euler totient, Dedekind psi, and sum of divisors functions evaluated at the largest integer function of x/n, extending understanding of their average orders.

Contribution

It offers a short elementary proof and improved error bounds for the asymptotics of sums involving these arithmetic functions, including first proofs for related sums.

Findings

Sharp asymptotic formulas with improved error terms for sums involving phi, psi, and sigma functions.

First proofs of asymptotics for sums of psi and sigma functions at largest integer.

Elementary proof techniques simplify previous complex arguments.

Abstract

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.