Vlad Robu: A generalisation of Euler's totient function, 413-418

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 $\varphi_P(n)$.

Key Words: Euler's totient function, Mertens' third theorem, irreducible integer polynomial, asymptotic lower bound, Frobenius density theorem, natural/Dirichlet density.

2020 Mathematics Subject Classification: Primary 11C08; Secondary 11N37.

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