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

TL;DR

This paper investigates the average number of ways to express an integer as a sum of polynomial values at prime powers, extending previous results to more general polynomials with specific degree and leading coefficient conditions.

Contribution

It generalizes earlier work by analyzing representations involving polynomials with integer coefficients, degree at least one, and leading coefficient one, for sums of prime power evaluations.

Findings

Extended results to polynomials with degree ≥ 1 and leading coefficient 1.

Analyzed the number of representations for sums involving these polynomials.

Built upon previous work focused on monomials and specific parameters.

Abstract

We study the average number of representations of an integer $n$ as $n = \phi(n_{1}) + \dots + \phi(n_{j})$, for polynomials $\phi \in \mathbb{Z}[n]$ with $\partial\phi = k\ge 1$, $\operatorname{lead}(\phi) = 1$, $j \ge k$, 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 $\phi(n) = n^k$, $k\ge 2$ and $j=k, k + 1$.