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

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 $X^{\frac{1}{2}+\epsilon }\le H<X^{1-\epsilon }$. In this article, we improve the range of $H$, that is, we prove the same asymptotic formula for $X^{\frac{32-4\sqrt{15}}{49}+\epsilon }\le H\le X^{1-\epsilon }$.