{"paper":{"title":"$\\sum_{p\\le n} 1/p = \\ln(\\ln n) + O(1)$: An Exposition","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.HO","authors_text":"Larry Washington, William Gasarch","submitted_at":"2015-11-05T17:34:32Z","abstract_excerpt":"It is well known that $\\sum_{p\\le n} 1/p =\\ln(\\ln(n)) + O(1)$ where $p$ goes over the primes. We give several known proofs of this.\n  We first present a a proof that $\\ge \\ln(\\ln(n)) + O(1)$. This is based on Euler's proof that $\\sum_p 1/p$ diverges. We then present three proofs that $\\sum_{p\\le n} 1/p \\le \\ln(\\ln(n)) + O(1)$ The first one, due to Mertens, does not use the prime number theorem. The second and third one do use the prime number theorem and hence are shorter."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.01823","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}