{"paper":{"title":"Inequalities involving Higher Degree Polynomial Functions in $\\pi(x)$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.GM","authors_text":"Subham De","submitted_at":"2024-07-24T07:49:17Z","abstract_excerpt":"The primary purpose of this article is to study the asymptotic and numerical estimates in detail for higher degree polynomials in $\\pi(x)$ having a general expression of the form, \\begin{align*} P(\\pi(x)) - \\frac{e x}{\\log x} Q(\\pi(x/e)) + R(x) \\end{align*} $P$, $Q$ and $R$ are arbitrarily chosen polynomials and $\\pi(x)$ denotes the \\textit{Prime Counting Function}. The proofs require specific order estimates involving $\\pi(x)$ and the \\textit{Second Chebyshev Function} $\\psi(x)$, as well as the famous \\textit{Prime Number Theorem} in addition to certain meromorphic properties of the \\textit{R"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2407.18983","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2407.18983/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"}