A Generalisation of Euler Totient Function

Euler’s totient function, $\phi (n)$, which counts how many of $0,1,\dots ,n-1$ are coprime to $n$, has an explicit asymptotic lower bound of $n\mathrm{/}\log \log n$, modulo some constant. In this note, we generalise $\phi$; given an irreducible integer polynomial $P$, we define the arithmetic function ${\phi }_P(n)$ 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 ${\phi }_P(n)$.