On the average number of representations of an integer as a sum of polynomials computed at prime values
Alessandra Migliaccio, Alessandro Zaccagnini
TL;DR
The paper addresses the problem of counting representations of integers as sums of polynomial values evaluated at prime powers, focusing on the average over short intervals. It extends prior work to general polynomials $\phi$ with $\partial\phi=k$ and leading coefficient $1$ (and later general $a_k$) for all $j\ge k$, deriving a precise asymptotic with explicit constants: $\sum_{n=N+1}^{N+H} R_{\phi,j}(n)=\frac{\gamma_k^j}{\gamma_{k,j}}HN^{(j-k)/k}+O\big(HN^{(j-k)/k}A(N;-c)\big)$, uniformly for $N^{1-13/(15k)+\varepsilon}<H<N^{1-\varepsilon}$. The method builds on exponential-sum techniques with a major/minor-arc decomposition centered on $\widetilde{S}_{\phi}(\alpha)$ and $U(\alpha,H)$, leveraging an $L^2$-estimate to control the minor arc and a Tolev-type bound for the error term, while Guth–Maynard results enhance the uniform range. The work thus generalizes and sharpens previous results, and provides a robust framework to handle leading coefficients other than $1$, with the main term adjusted by a factor $a_k^{-j/k}$. Overall, the results deepen our understanding of average representations by polynomial values at prime powers and illustrate how modern zero-density results can improve uniformity ranges.
Abstract
We study the average number of representations of an integer $n$ as $n = φ(n_{1}) + \dots + φ(n_{j})$, for polynomials $φ\in \mathbb{Z}[n]$ with $\partialφ= k\ge 1$, $\operatorname{lead}(φ) = 1$, $j \ge k$, where $n_{i}$ is a prime power for each $i \in \{1, \dots, j\}$. We extend the results of Languasco and Zaccagnini (2019), for $k=3$ and $j=4$, and of Cantarini, Gambini and Zaccagnini (2020), where they focused on monomials $φ(n) = n^k$, $k\ge 2$ and $j=k, k + 1$.