Large gaps between values of several binary quadratic forms
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866918144135659520 |
|---|---|
| author | Żmija, Błażej |
| author_facet | Żmija, Błażej |
| contents | In this paper we study the problem of long gaps between values of binary quadratic forms. Let $D_{1}$, $D_{2},\ldots ,D_{r}$ be negative integers and $(s_{n})_{n=1}^{\infty}$ be the sequence of all the numbers representable by any binary quadratic form of discriminant $D_{1}$, $D_{2}$, $\ldots$ or $D_{r}$, and let $d :={\rm lcm}\{D_{1},\ldots ,D_{r}\}$. We show that then \begin{align*} \limsup_{n\to\infty}\frac{s_{n+1}-s_{n}}{\log s_{n}}\geq \frac{1}{\log d + \log\log d + \log\log\log d + 4}. \end{align*} This improves and generalises a result by Dietmann, Elsholtz, Kalmynin, Konyagin, and Maynard.
As a by-product of our preliminary results, we show an improvement to the Pólya-Vinogradov inequality. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_15365 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Large gaps between values of several binary quadratic forms Żmija, Błażej Number Theory 11N25, 11N37 In this paper we study the problem of long gaps between values of binary quadratic forms. Let $D_{1}$, $D_{2},\ldots ,D_{r}$ be negative integers and $(s_{n})_{n=1}^{\infty}$ be the sequence of all the numbers representable by any binary quadratic form of discriminant $D_{1}$, $D_{2}$, $\ldots$ or $D_{r}$, and let $d :={\rm lcm}\{D_{1},\ldots ,D_{r}\}$. We show that then \begin{align*} \limsup_{n\to\infty}\frac{s_{n+1}-s_{n}}{\log s_{n}}\geq \frac{1}{\log d + \log\log d + \log\log\log d + 4}. \end{align*} This improves and generalises a result by Dietmann, Elsholtz, Kalmynin, Konyagin, and Maynard. As a by-product of our preliminary results, we show an improvement to the Pólya-Vinogradov inequality. |
| title | Large gaps between values of several binary quadratic forms |
| topic | Number Theory 11N25, 11N37 |
| url | https://arxiv.org/abs/2509.15365 |