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arXiv · 2505.03447

Asymptotic formula for the sum of a prime and a square-full number in short intervals shorter than $X^{1/2}$

Fumi Ogihara
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Abstract

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 $X^{\frac{32-4\sqrt{15}}{49}+\varepsilon} \le H \le X^{1- \varepsilon}$.

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Fumi Ogihara. 2025-05-06. Asymptotic formula for the sum of a prime and a square-full number in short intervals shorter than $X^{1/2}$. https://arxiv.org/abs/2505.03447

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