The variance of the Euler totient function

Abstract

In this paper we study the variance of the Euler totient function (normalized to $\phi (n)/n$) 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 ${\mathbb{F}}_q[T]$, $q\to \infty$, the analogue does not hold: due to a high amount of cancellation, the variance becomes inversely proportional to the size of the interval.

15 pages, 3 figures

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