Asymptotic formula for the sum of a prime and a square-full number in short intervals

TL;DR

This paper derives an asymptotic formula for counting numbers that are sums of a prime and a square-full number within short intervals slightly larger than the square root of the starting point.

Contribution

It provides the first asymptotic estimate for the sum of representations of integers as a prime plus a square-full number in short intervals.

Findings

Established an asymptotic formula for the sum over short intervals.

Extended understanding of additive representations involving primes and square-full numbers.

Demonstrated the formula's validity for intervals slightly larger than the square root of the starting point.

Abstract

Let $R_{m, \mathrm{sq-full}}(N)$ be a representation function for the sum of a prime and a square-full number. In this article, we prove an asymptotic formula for the sum of $R_{m, \mathrm{sq-full}}(N)$ over positive integers $N$ in a short interval ($X$, $X+H$] of length $H$ slightly bigger than $X^{\frac{1}{2}}$.