On divisor bounded multiplicative functions in short intervals
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916286650384384 |
|---|---|
| author | Sun, Yu-Chen |
| author_facet | Sun, Yu-Chen |
| contents | Let $d_k(n) = \sum_{n_1 \cdots n_k = n}1$ be the $k$-fold divisor function. We call a function $f:\mathbb{N} \to \mathbb{C}$ a $d_k$-bounded multiplicative function, if $f$ is multiplicative and $|f(n)| \leq d_k(n)$ for every $n \in \mathbb{N}$. In this paper we improve Mangerel's results which extend the Matomäki-Radziwi{łł} theorem to divisor bounded multiplicative functions.
In particular, we prove that for sufficiently large $X \geq 2$, any $ε>0$ and $h \geq (\log X)^{k \log k - k + 1 + ε}$ , we have
$$
\frac{1}{h}\sum_{x<n\leq x+h}d_k(n)-\frac{1}{x}\sum_{x<n\leq 2x}d_{k}(n) = o(\log^{k-1} x)
$$
for almost all $x \in [X,2X]$. We also demonstrate that the exponent $k \log k-k+1$ is optimal. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_08432 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On divisor bounded multiplicative functions in short intervals Sun, Yu-Chen Number Theory Let $d_k(n) = \sum_{n_1 \cdots n_k = n}1$ be the $k$-fold divisor function. We call a function $f:\mathbb{N} \to \mathbb{C}$ a $d_k$-bounded multiplicative function, if $f$ is multiplicative and $|f(n)| \leq d_k(n)$ for every $n \in \mathbb{N}$. In this paper we improve Mangerel's results which extend the Matomäki-Radziwi{łł} theorem to divisor bounded multiplicative functions. In particular, we prove that for sufficiently large $X \geq 2$, any $ε>0$ and $h \geq (\log X)^{k \log k - k + 1 + ε}$ , we have $$ \frac{1}{h}\sum_{x<n\leq x+h}d_k(n)-\frac{1}{x}\sum_{x<n\leq 2x}d_{k}(n) = o(\log^{k-1} x) $$ for almost all $x \in [X,2X]$. We also demonstrate that the exponent $k \log k-k+1$ is optimal. |
| title | On divisor bounded multiplicative functions in short intervals |
| topic | Number Theory |
| url | https://arxiv.org/abs/2401.08432 |