A Generalisation of Euler Totient Function

2022/11/19 by Vlad Robu, Robu, Vlad
Mathematics · #11C08 (Primary) #11N37 (Secondary) #Advanced Mathematical Identities #Advanced Mathematical Theories #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2211.10644

openalex publication_date 2022/11/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Euler's totient function, φ(n), which counts how many of 0,1,…,n-1 are coprime to n, has an explicit asymptotic lower bound of n/log log n, modulo some constant. In this note, we generalise φ; given an irreducible integer polynomial P, we define the arithmetic function φ_P(n) that counts the amount of numbers among P(0),P(1),…,P(n-1) that are coprime to n. We also provide an asymptotic lower bound for φ_P(n).

Related

• Inequalities involving the primorial counting function
• Explicit formulas for Euler's totient function and the number of divisors
• Some remarks on sum of Euler's totient function
• Counting invertible sums of squares modulo n and a new generalization of Euler totient function
• Counting integers with a smooth totient