Variance of square-full integers in short intervals and arithmetic progressions
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908515958784000 |
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| author | Meemark, Yotsanan Wongcharoenbhorn, Watcharakiete |
| author_facet | Meemark, Yotsanan Wongcharoenbhorn, Watcharakiete |
| contents | Define a natural number $n$ as a \textit{square-full} integer if for every prime $p$ such that $p|n$, we have $p^2|n$. In this paper, we establish an upper bound on the variance of square-full integers in short intervals of an expected order, under the assumption of a certain quasi-Riemann hypothesis. We also prove an asymptotic formula for the variance in arithmetic progressions, averaging over a quadratic residue and a nonresidue by a half, which is of smaller order of magnitude than the aforementioned bound for all primes $q\gg x^{51/114+\varepsilon}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_14511 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Variance of square-full integers in short intervals and arithmetic progressions Meemark, Yotsanan Wongcharoenbhorn, Watcharakiete Number Theory Define a natural number $n$ as a \textit{square-full} integer if for every prime $p$ such that $p|n$, we have $p^2|n$. In this paper, we establish an upper bound on the variance of square-full integers in short intervals of an expected order, under the assumption of a certain quasi-Riemann hypothesis. We also prove an asymptotic formula for the variance in arithmetic progressions, averaging over a quadratic residue and a nonresidue by a half, which is of smaller order of magnitude than the aforementioned bound for all primes $q\gg x^{51/114+\varepsilon}$. |
| title | Variance of square-full integers in short intervals and arithmetic progressions |
| topic | Number Theory |
| url | https://arxiv.org/abs/2504.14511 |