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

2026/01/30 by Alessandra Migliaccio, Alessandro Zaccagnini
Mathematics · #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #Analytic Number Theory Research #math.NT

paper · pdf · doi:10.1016/j.jnt.2026.06.012

openalex publication_date 2026/07/24 · openalex created_date 2026/07/25 · openalex updated_date 2026/07/30

Abstract

We study the average number of representations of an integer n as n = ϕ(n₁) + … + ϕ(n_j), for polynomials ϕ∈ ℤ[n] with ∂ϕ= k≥ 1, lead(ϕ) = 1, j ≥ k, where n_i is a prime power for each i ∈ \1, …, 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≥ 2 and j=k, k + 1.

Citations

• New large value estimates for Dirichlet polynomials
• On the average number of representationsof an integer as a sum of like prime powers
• A Diophantine approximation problem with two primes and one k-th power of a prime
• Sum of one prime and two squares of primes in short intervals
• Hilbert's Inequality

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