Weighted averages of arithmetic functions and applications to equidistribution and ergodic theory
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2026
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866913177424363520 |
|---|---|
| author | Bergelson, Vitaly Reilly, Michael Richter, Florian K. |
| author_facet | Bergelson, Vitaly Reilly, Michael Richter, Florian K. |
| contents | 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. (For instance, one may take $\vartheta(n)=Ω(n)$, where $Ω(n)$ counts the number of prime factors of $n$ with multiplicity, or $\vartheta(n)=s_q(p_n)$, where $s_q$ is the sum-of-digits function in base $q$ and $p_n$ denotes the $n$-th prime. Additional natural examples are discussed in the paper.) Building on 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 several novel applications. For example, it follows that $(Ω(n)^c)_{n\in\mathbb{N}}$ is uniformly distributed mod $1$ if and only if $c$ is a non-integer greater than $\frac{1}{2}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_07186 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Weighted averages of arithmetic functions and applications to equidistribution and ergodic theory Bergelson, Vitaly Reilly, Michael Richter, Florian K. Number Theory 11K06, 11N37, 11K65, 11N64 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. (For instance, one may take $\vartheta(n)=Ω(n)$, where $Ω(n)$ counts the number of prime factors of $n$ with multiplicity, or $\vartheta(n)=s_q(p_n)$, where $s_q$ is the sum-of-digits function in base $q$ and $p_n$ denotes the $n$-th prime. Additional natural examples are discussed in the paper.) Building on 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 several novel applications. For example, it follows that $(Ω(n)^c)_{n\in\mathbb{N}}$ is uniformly distributed mod $1$ if and only if $c$ is a non-integer greater than $\frac{1}{2}$. |
| title | Weighted averages of arithmetic functions and applications to equidistribution and ergodic theory |
| topic | Number Theory 11K06, 11N37, 11K65, 11N64 |
| url | https://arxiv.org/abs/2604.07186 |