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

TL;DR

This paper extends the range of short intervals for which an asymptotic formula accurately counts representations of numbers as a sum of a prime and a square-full number, improving previous results in analytic number theory.

Contribution

The authors improve the interval length range for which the asymptotic formula for the sum of representations as prime plus square-full number holds.

Findings

Asymptotic formula valid for shorter intervals than before

Extended the lower bound of interval length to approximately $X^{0.519}$

Confirmed the asymptotic behavior in a broader range of short intervals

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