Distribution of values of general Euler totient function

2023/04/05 by Debika Banerjee, Banerjee, Debika, Bittu Chahal +5
Mathematics · #11M06 #11N60 #11N64 #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Meromorphic and Entire Functions #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2304.02540

openalex publication_date 2023/04/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let Φ_k(n)=|\ (x₁, x₂, ⋯, x_k)∈ (ℤ/nℤ)^k; gcd(x₁²+x₂²+ ⋯+ x_k², n)=1\| be a general totient function introduced first by Caldéron et. al. Motivated by the classical works of Schoenberg, Erdős, Bateman and Diamond on the distribution of Φ₁(n), we prove results on the joint distribution of Φ_k(n) for any k≥ 1. Additionally, we also exhibit the extremal order of Φ_k(n).

Related

• The distribution of gcd(n,ϕ(n))
• An improved upper bound for the distribution of iterated Euler totient functions
• The distribution functions of σ(n)/n and n/ϕ(n), II
• Euler's ℓ-totients and Riemann hypothesis
• A Generalisation of Euler Totient Function