On the average number of representations of an integer as a sum of polynomials computed at prime values

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 $j=k, k + 1$.