{"paper":{"title":"A Dynamical Criterion Equivalent to the Riemann Hypothesis","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.GM","authors_text":"Hendrik Wladimir Albrecht Edwin Kuipers","submitted_at":"2025-09-12T08:59:19Z","abstract_excerpt":"We introduce a discrete dynamical system on the integers, defined by moving a composite $m$ forward to $m+\\pi(m)$ and a prime $p$ backward to $p-\\mathrm{prevprime}(p)$. This map produces trajectories whose contraction properties are closely tied to the distribution of primes.\n  We prove unconditional contraction inequalities for error terms derived from these trajectories, using explicit remainder bounds from the smoothed explicit formula. Building on this, we show that the Riemann Hypothesis is equivalent to a sharp contraction condition: the trajectory error functional satisfies $E(X)\\ll X^{"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2509.10588","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/2509.10588/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"}