The variance of the Euler totient function

TL;DR

This paper investigates the variance of the normalized Euler totient function in integers and polynomial rings over finite fields, revealing constant variance in integers and inverse proportionality in polynomial rings as the field size grows.

Contribution

It provides new insights into the behavior of the Euler totient function's variance in different algebraic settings, highlighting contrasting phenomena.

Findings

Variance in integers tends to a constant under certain conditions.

In polynomial rings over finite fields, variance decreases inversely with interval size.

Numerical simulations support the theoretical results.

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.