Square-full values of quadratic polynomials
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866918286783938560 |
|---|---|
| author | Wongcharoenbhorn, Watcharakiete Meemark, Yotsanan |
| author_facet | Wongcharoenbhorn, Watcharakiete Meemark, Yotsanan |
| contents | A $\textit{square-full}$ number is a positive integer for which all its prime divisors divide itself at least twice. The counting function of square-full integers of the form $f(n)$ for $n\leqslant N$ is denoted by $S^{{\mathstrut\hspace{0.05em}\blacksquare}}_f(N)$. We have known that for a relatively prime pair $(a,b)\in\mathbb N\times \mathbb N\cup\{0\}$ with a linear polynomial $f(x)=ax+b$, its counting function is $\asymp_{a,b} N^\frac{1}{2}$. Fix $\varepsilon>0$, for an admissible quadratic polynomial $f(x)$, we prove that $$S^{{\mathstrut\hspace{0.05em}\blacksquare}}_f(N)\ll_{\varepsilon, f} N^{\varpi+\varepsilon}$$ for some absolute constant $\varpi<1/2$. Under the assumption on the $abc$ conjecture, we expect the upper bound to be $O_{\varepsilon,f}(N^\varepsilon)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_06968 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Square-full values of quadratic polynomials Wongcharoenbhorn, Watcharakiete Meemark, Yotsanan Number Theory Primary 11N32, Secondary 11D45 A $\textit{square-full}$ number is a positive integer for which all its prime divisors divide itself at least twice. The counting function of square-full integers of the form $f(n)$ for $n\leqslant N$ is denoted by $S^{{\mathstrut\hspace{0.05em}\blacksquare}}_f(N)$. We have known that for a relatively prime pair $(a,b)\in\mathbb N\times \mathbb N\cup\{0\}$ with a linear polynomial $f(x)=ax+b$, its counting function is $\asymp_{a,b} N^\frac{1}{2}$. Fix $\varepsilon>0$, for an admissible quadratic polynomial $f(x)$, we prove that $$S^{{\mathstrut\hspace{0.05em}\blacksquare}}_f(N)\ll_{\varepsilon, f} N^{\varpi+\varepsilon}$$ for some absolute constant $\varpi<1/2$. Under the assumption on the $abc$ conjecture, we expect the upper bound to be $O_{\varepsilon,f}(N^\varepsilon)$. |
| title | Square-full values of quadratic polynomials |
| topic | Number Theory Primary 11N32, Secondary 11D45 |
| url | https://arxiv.org/abs/2405.06968 |