Weighted averages of arithmetic functions and applications to equidistribution

TL;DR

This paper develops a general method for estimating weighted averages of arithmetic functions and applies it to study the uniform distribution of sequences derived from prime factor counts and Hardy field functions.

Contribution

It introduces a broad theorem for weighted averages of arithmetic functions satisfying Gaussian distribution conditions, with applications to uniform distribution problems.

Findings

Sequences like ig(\u03a9(n)) are uniformly distributed mod 1 if and only if the exponent is a non-integer greater than 1/2.

The main theorem applies to functions ig(h(\u03a9(n))) with Hardy field functions, characterizing their distribution.

New results on the distribution of sequences involving ig(ig( ig)), ig(ig( ig)), and ig(ig( ig)).

Abstract

For a wide range of functions $W\colon\mathbb{N}\to\mathbb{N}$, we establish a general result for estimating weighted averages of the form \[ \mathbb{E}^{W}_{n \le N} f(\vartheta(n))= \frac{1}{W(N)}\sum_{n=1}^N (W(n)-W(n-1))f(\vartheta(n)), \] where $f\colon \{1,\ldots,N\}\to\mathbb{C}$ is an arbitrary function, and $\vartheta(n)$ is any arithmetic function that adheres to a certain Gaussian distribution condition. (In particular, one can take $\vartheta(n)=\Omega(n)$, $\vartheta(n)=\omega(n)$, or $\vartheta(n)=\Omega(q_n)$, where $\Omega(n)$ and $\omega(n)$ count the number of prime factors of $n$ with and without multiplicities respectively, and $q_n$ denotes the $n$-th squarefree number.) As an application of our main theorem, we show that if $h(n)$ is a function from a Hardy field with polynomial growth then $(h(\vartheta(n)))_{n\in\mathbb{N}}$ is uniformly distributed mod $1$ if and only if one of the following (mutually exclusive) conditions is satisfied: (i) $\lim_{x\to\infty} \frac{|h(x)-p(x)|}{x \log x}=\infty$ for all $p(x)\in \mathbb{Q}[x]$; (ii) $\lim_{x\to\infty}\frac{|h(x)-p(x)|}{\sqrt{x}}=\infty$ for each $p(x)\in \mathbb{Q}[x]$ and there exists $q(x)\in \mathbb{Q}[x]$ such that $\lim_{x\to\infty}\frac{|h(x)-q(x)|}{x}<\infty$. This leads to novel applications regarding the uniform distribution of sequences of the from $h(\Omega(n))$, $h(\omega(n))$, and $h(\Omega(q_n))$. For example, we show that $(\Omega(n)^c)_{n\in\mathbb{N}}$ is uniformly distributed mod $1$ if and only if $c$ is a non-integer greater than $\frac{1}{2}$.