A Generalisation of Euler Totient Function

TL;DR

This paper generalizes Euler's totient function to polynomials, defining a new arithmetic function that counts coprime values of polynomial sequences and establishing its asymptotic lower bound.

Contribution

It introduces a generalized totient function for polynomial sequences and derives an asymptotic lower bound, extending classical number theory results.

Findings

Defined $_P(n)$ for irreducible polynomials $P$

Established asymptotic lower bound for $_P(n)$

Extended classical totient function properties to polynomial sequences

Abstract

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