Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2406.19788 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909233002315776 |
|---|---|
| author | Ye, Zhaoxi Xu, Zhefeng |
| author_facet | Ye, Zhaoxi Xu, Zhefeng |
| contents | Let $p$ be a prime number, $k\ge 0$ and $f$ be a class of arithmetic functions satisfying some simple conditions. In this short paper, we study the asymptotical behaviour of summation function $$ψ_{f,k}(x):=\sum_{n\le x}Λ(n)\frac{f\left ( \left [ \frac{x}{n} \right ] \right ) }{\left [ \frac{x}{n} \right ]^{k} } ,~~~~~~~~~~~ π_{f,k}(x):=\sum_{p\le x}\frac{f\left ( \left [ \frac{x}{p} \right ] \right ) }{\left [ \frac{x}{p} \right ]^{k} } $$ as $x\to \infty $, where $\left [ \cdot \right ] $ is the integral part function, $Λ(n)$ is the von Mangoldt function. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_19788 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Involves averaging arithmetic and integral partial functions over sparse set Ye, Zhaoxi Xu, Zhefeng Number Theory Let $p$ be a prime number, $k\ge 0$ and $f$ be a class of arithmetic functions satisfying some simple conditions. In this short paper, we study the asymptotical behaviour of summation function $$ψ_{f,k}(x):=\sum_{n\le x}Λ(n)\frac{f\left ( \left [ \frac{x}{n} \right ] \right ) }{\left [ \frac{x}{n} \right ]^{k} } ,~~~~~~~~~~~ π_{f,k}(x):=\sum_{p\le x}\frac{f\left ( \left [ \frac{x}{p} \right ] \right ) }{\left [ \frac{x}{p} \right ]^{k} } $$ as $x\to \infty $, where $\left [ \cdot \right ] $ is the integral part function, $Λ(n)$ is the von Mangoldt function. |
| title | Involves averaging arithmetic and integral partial functions over sparse set |
| topic | Number Theory |
| url | https://arxiv.org/abs/2406.19788 |