{"paper":{"title":"Power Partitions and Hayman Functions","license":"http://creativecommons.org/licenses/by/4.0/","headline":"The generating function for partitions into k-th powers is strongly Gaussian in the Báez-Duarte sense.","cross_cats":["math.CV","math.NT"],"primary_cat":"math.PR","authors_text":"Jos\\'e L. Fern\\'andez, V\\'ictor J. Maci\\'a","submitted_at":"2026-02-20T19:32:11Z","abstract_excerpt":"We prove, within the probabilistic framework of Khinchin families, that the generating function $P_k$ of partitions into $k$-th powers is strongly Gaussian in the sense of B\\'aez-Duarte, and even further that it is a Hayman function. Thus the Hardy--Ramanujan asymptotic formula for the number $p_k(n)$ of partitions of $n$ into $k$-th powers which reads \\[ p_k(n) \\sim \\frac{\\alpha_k}{n^{(3k+1)/(2k+2)}} \\exp\\!\\Big(\\beta_k\\, n^{1/(k+1)}\\Big), \\qquad n\\to\\infty, \\] where $\\alpha_k$ and~$\\beta_k$ are explicit constants depending only on $k$, follows directly from Hayman's asymptotic formula for str"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We prove that the generating function of partitions into k-th powers is strongly Gaussian in the sense of Báez-Duarte.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The bounds of Tenenbaum, Wu and Li on the generating function are strong enough to verify the Gaussianity criterion for Khinchin families.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"The generating function of k-th power partitions is strongly Gaussian, so the asymptotic p_k(n) ~ alpha_k n^(-(3k+1)/(2k+2)) exp(beta_k n^{1/(k+1)}) follows from Hayman's theorem via mean and variance approximations.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The generating function for partitions into k-th powers is strongly Gaussian in the Báez-Duarte sense.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"bbb805684aae87e1c00f679dc807cfd45edd608a9b107d9768c5b8065c9f7926"},"source":{"id":"2602.18575","kind":"arxiv","version":3},"verdict":{"id":"6b2f1442-732c-498d-96df-f4c28863a483","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T20:32:06.864710Z","strongest_claim":"We prove that the generating function of partitions into k-th powers is strongly Gaussian in the sense of Báez-Duarte.","one_line_summary":"The generating function of k-th power partitions is strongly Gaussian, so the asymptotic p_k(n) ~ alpha_k n^(-(3k+1)/(2k+2)) exp(beta_k n^{1/(k+1)}) follows from Hayman's theorem via mean and variance approximations.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The bounds of Tenenbaum, Wu and Li on the generating function are strong enough to verify the Gaussianity criterion for Khinchin families.","pith_extraction_headline":"The generating function for partitions into k-th powers is strongly Gaussian in the Báez-Duarte sense."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2602.18575/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}