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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 $j=k, k + 1$."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2601.22822","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.NT","submitted_at":"2026-01-30T10:47:09Z","cross_cats_sorted":[],"title_canon_sha256":"bb0be3600fa6e60819f4d3012112a1ea30be0f10b810b9845effe71ba5f1c17a","abstract_canon_sha256":"40ac21c7e8d49a777b4010abef68405c930e232ab5b6118afb5339426a53260b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:45:05.667869Z","signature_b64":"S2oDC1ww3VXS71IhpoywGjhEtFk/AAEHpqSw4xygihE5txTpTtBMcpTPnmogM0r36YATPkd/PbiFvwggXGvnBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3e1d6b11f8564c56f0fd7cbf87a85adee84639cb78e44078aa6739203f9877fc","last_reissued_at":"2026-05-18T02:45:05.667209Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:45:05.667209Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 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