The variance of the Euler totient function

Abstract

In this paper we study the variance of the Euler totient function (normalized to $\varphi(n)/n$) in the integers $\mathbb{Z}$ and in the polynomial ring $\mathbb{F}_q[T]$ over a finite field $\mathbb{F}_q$. It turns out that in $\mathbb{Z}$, under some assumptions, the variance of the normalized Euler function becomes constant. This is supported by several numerical simulations. Surprisingly, in $\mathbb{F}_q[T]$, $q\rightarrow \infty$, the analogue does not hold: due to a high amount of cancellation, the variance becomes inversely proportional to the size of the interval.