On the sum of a prime power and a power in short intervals

TL;DR

This paper extends previous results on the average number of representations of numbers as a sum of a prime power and a positive integer power within short intervals, improving the interval length for which asymptotic formulas hold.

Contribution

It proves that the asymptotic formula for the representation function holds for shorter intervals of size approximately $X^{0.337}$, extending prior results to a broader range.

Findings

Asymptotic formula holds for intervals of size $X^{0.337}$

Generalizes results to arbitrary $(k, \, ext{ell})$ cases

Provides unconditional results for $\\ell=2$ case

Abstract

Let $R_{k,\ell}(N)$ be the representation function for the sum of the $k$-th power of a prime and the $\ell$-th power of a positive integer. Languasco and Zaccagnini (2017) proved an asymptotic formula for the average of $R_{1,2}(N)$ over short intervals $(X,X+H]$ of the length $H$ slightly shorter than $X^{\frac{1}{2}}$, which is shorter than the length $H=X^{\frac{1}{2}+\epsilon}$ in the exceptional set estimates of Mikawa (1993) and of Perelli and Pintz (1995). In this paper, we prove that the same asymptotic formula for $R_{1,2}(N)$ holds for $H$ of the size $X^{0.337}$. Recently, Languasco and Zaccagnini (2018) extended their result to more general $(k,\ell)$. We also consider this general case, and as a corollary, we prove a conditional result of Languasco and Zaccagnini (2018) for the case $\ell=2$ unconditionally up to some small factors.