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

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 $X^{\frac{1}{2}+\varepsilon}$ 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.

Keywords

Waring–Goldbach problem, exponential sums, zero density estimates

2010 Mathematics Subject Classification

Primary 11P32. Secondary 11L07.

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This work was supported by Grant-in-Aid for JSPS Research Fellow Grant Number JP16J00906 and JSPS KAKENHI Grant Numbers 19K23402, 21K13772.

Received 13 June 2019

Received revised 13 July 2022

Accepted 24 October 2022

Published 13 November 2023