{"paper":{"title":"On the arithmetic average of the first $n$ primes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Matt Visser (Victoria University of Wellington)","submitted_at":"2025-05-08T05:10:43Z","abstract_excerpt":"The arithmetic average of the first $n$ primes, $\\bar p_n = {1\\over n} \\sum_{i=1}^n p_i$, exhibits very many interesting and subtle properties. Since the transformation from $p_n \\to \\bar p_n$ is extremely easy to invert, $p_n = n\\bar p_n - (n-1)\\bar p_{n-1}$, it is clear that these two sequences $p_n \\longleftrightarrow \\bar p_n$ must ultimately carry exactly the same information. But the averaged sequence $\\bar p_n$, while very closely correlated with the primes, ($\\bar p_n \\sim {1\\over2} p_n$), is much \"smoother'', and much better behaved. Using extensions of various standard results I shal"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2505.04951","kind":"arxiv","version":3},"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/2505.04951/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"}