{"paper":{"title":"Prime Running Functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Jaeyoon Kim","submitted_at":"2020-06-23T21:57:02Z","abstract_excerpt":"We study arithmetic functions $\\Phi(x;d,a)$, called prime running functions, whose value at $x$ sums the gaps between primes $p_k \\equiv a\\ (\\text{mod}\\ d)$ below $x$ and the next following prime $p_{k+1}$, up to $x$. (The following prime $p_{k+1}$ may be in any residue class $(\\text{mod}\\ d)$.) We empirically observe systematic biases of order $x / \\log x$ in $\\Phi(x;d,a) - \\Phi(x;d,b)$ for different $a,b$. We formulate modified Cram\\'er models for primes and show that the corresponding sum of prime gap statistics exhibits systematic biases of this order of magnitude. The predictions of such "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2006.13355","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/2006.13355/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"}