On the average number of representations of an integer as a sum of polynomials computed at prime values
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866914562244083712 |
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| author | Migliaccio, Alessandra Zaccagnini, Alessandro |
| author_facet | Migliaccio, Alessandra Zaccagnini, Alessandro |
| contents | 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 $j=k, k + 1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_22822 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the average number of representations of an integer as a sum of polynomials computed at prime values Migliaccio, Alessandra Zaccagnini, Alessandro Number Theory Primary 11P32. Secondary 11P55, 11P05 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 $j=k, k + 1$. |
| title | On the average number of representations of an integer as a sum of polynomials computed at prime values |
| topic | Number Theory Primary 11P32. Secondary 11P55, 11P05 |
| url | https://arxiv.org/abs/2601.22822 |