{"paper":{"title":"On the average number of representations of an integer as a sum of polynomials computed at prime values","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"The average number of representations of an integer as a sum of j values of a degree-k polynomial at prime powers follows a positive asymptotic for j at least k.","cross_cats":[],"primary_cat":"math.NT","authors_text":"Alessandra Migliaccio, Alessandro Zaccagnini","submitted_at":"2026-01-30T10:47:09Z","abstract_excerpt":"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 $j=k, k + 1$."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We study the average number of representations of an integer n as n = φ(n1) + … + φ(nj), for polynomials φ ∈ ℤ[n] with ∂φ = k ≥ 1, lead(φ) = 1, j ≥ k, where ni is a prime power for each i.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the analytic machinery (likely circle method or Vinogradov-type estimates) from the cited prior works extends uniformly to arbitrary k and all j ≥ k without additional restrictions on the polynomial or error terms.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Asymptotic formulas are established for the average number of ways to write n as sum of j monic polynomial values at prime powers, generalizing earlier cases for cubics and monomials.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The average number of representations of an integer as a sum of j values of a degree-k polynomial at prime powers follows a positive asymptotic for j at least k.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"08e97b2f0c4ea66a84fa19d73f5d34c0097b7437587724e5d69454ddcbeb67e2"},"source":{"id":"2601.22822","kind":"arxiv","version":2},"verdict":{"id":"9bdbfbce-2aa6-41f0-8191-c916423476cd","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-16T09:39:37.534738Z","strongest_claim":"We study the average number of representations of an integer n as n = φ(n1) + … + φ(nj), for polynomials φ ∈ ℤ[n] with ∂φ = k ≥ 1, lead(φ) = 1, j ≥ k, where ni is a prime power for each i.","one_line_summary":"Asymptotic formulas are established for the average number of ways to write n as sum of j monic polynomial values at prime powers, generalizing earlier cases for cubics and monomials.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the analytic machinery (likely circle method or Vinogradov-type estimates) from the cited prior works extends uniformly to arbitrary k and all j ≥ k without additional restrictions on the polynomial or error terms.","pith_extraction_headline":"The average number of representations of an integer as a sum of j values of a degree-k polynomial at prime powers follows a positive asymptotic for j at least k."},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}