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

2021/01/06 by N. A. Carella, Carella, N. A.
Mathematics · #2020: Primary 11N37 #Advanced Mathematical Identities #Advanced Mathematical Theories #Analytic Number Theory Research #FOS: Mathematics #General Mathematics (math.GM) #Secondary 11N05

paper · pdf · doi:10.48550/arxiv.2101.02248

openalex publication_date 2021/01/06 · openalex created_date 2021/06/07 · openalex updated_date 2026/07/28

Abstract

Let x≥ 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 ∑_{n≤ x}φ([x/n])=(6/π²)xlog x+O ( x(log x)^2/3(loglog x)^1/3 ) was proved very recently. This note presents a short elementary proof, and sharpen the error term to ∑_{n≤ x}φ([x/n])=(6/π²)xlog x+O(x) . In addition, the first proofs of the asymptotics formulas for the finite sums ∑_{n≤ x}ψ([x/n])=(15/π²)xlog x+O(xlog log x) , and ∑_{n≤ x}σ([x/n])=(π²/6)xlog x+O(x log log x) are also evaluated here.

Related

• Euler Totient Function And The Largest Integer Function Over The Shifted Primes
• Sum of Divisors Function And The Largest Integer Function Over The Shifted Primes
• Note on a sum involving the Euler function
• Arithmetic and Asymptotic Properties of Restricted Totient Sums
• The Error Term of the Summatory Euler Phi Function