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 $n=\varphi (n_1)+\cdots +\varphi (n_j)$, for polynomials $\varphi \in \mathbb{Z}[n]$ with $\partial \varphi =k\ge 1$, $\mathrm{lead}(\varphi )=1$, $j\ge k$, 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 $\varphi (n)=n^k$, $k\ge 2$ and $j=k,k+1$.

Final version. To appear in Journal of Number Theory