{"paper":{"title":"A Regularised Wallis Hierarchy","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"S. R. Holcombe","submitted_at":"2026-06-22T22:00:55Z","abstract_excerpt":"A hierarchy of regularised Wallis products is introduced by raising the reciprocal Wallis factor \\[ 1-\\frac1{n^2} \\] to the polynomial weight $n^m$, $m=0,1,2,\\ldots$. For each $m$, a minimal exponential counterterm is chosen by cancelling precisely the non-summable terms in the logarithmic expansion. This gives a convergent product $P_m$ the logarithm of which is an explicit zeta-function tail. The first non-trivial examples are \\[ \\prod_{n=2}^{\\infty} e^{1/n} \\left(1-\\frac1{n^2}\\right)^n = \\frac{e^\\gamma}{2}, \\qquad \\prod_{n=2}^{\\infty} e\\left(1-\\frac1{n^2}\\right)^{n^2} = \\frac{\\pi}{e^{3/2}}."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.23973","kind":"arxiv","version":1},"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/2606.23973/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"}