Asymptotic formula for the sum of a prime and a square-full number in short intervals shorter than $X^{1/2}$
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866912414035869696 |
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| author | Ogihara, Fumi |
| author_facet | Ogihara, Fumi |
| contents | Let $R(N)$ be the number of representations of $N$ as a sum of a prime and a square-full number weighted with logarithmic function. In $2024$, the author and Y. Suzuki obtained an asymptotic formula for the sum of $R(N)$ over positive integers $N$ in a short interval ($X$, $X+H$] for $X^{\frac{1}{2}+\varepsilon} \le H < X^{1-\varepsilon}$. In this article, we improve the range of $H$, that is, we prove the same asymptotic formula for $X^{\frac{32-4\sqrt{15}}{49}+\varepsilon} \le H \le X^{1- \varepsilon}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_03447 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Asymptotic formula for the sum of a prime and a square-full number in short intervals shorter than $X^{1/2}$ Ogihara, Fumi Number Theory 11P32 Let $R(N)$ be the number of representations of $N$ as a sum of a prime and a square-full number weighted with logarithmic function. In $2024$, the author and Y. Suzuki obtained an asymptotic formula for the sum of $R(N)$ over positive integers $N$ in a short interval ($X$, $X+H$] for $X^{\frac{1}{2}+\varepsilon} \le H < X^{1-\varepsilon}$. In this article, we improve the range of $H$, that is, we prove the same asymptotic formula for $X^{\frac{32-4\sqrt{15}}{49}+\varepsilon} \le H \le X^{1- \varepsilon}$. |
| title | Asymptotic formula for the sum of a prime and a square-full number in short intervals shorter than $X^{1/2}$ |
| topic | Number Theory 11P32 |
| url | https://arxiv.org/abs/2505.03447 |