{"paper":{"title":"Negative moments of the gaps between consecutive primes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Marek Wolf","submitted_at":"2018-05-13T20:13:50Z","abstract_excerpt":"We derive heuristically approximate formulas for the negative $k$--moments $M_{-k}(x)$ of the gaps between consecutive primes$<x $ represented directly by $\\pi(x)$ --- the number of primes up to $x$. In particular we propose an analytical formula for the sum of reciprocals of gaps between consecutive primes $<x : ~ M_{-1}(x)\\sim \\frac{\\pi^2(x)}{x-2\\pi(x)}\\log\\Big(\\frac{x}{2\\pi(x)}\\Big) \\sim x \\log \\log(x)/\\log^2(x)$. We illustrate obtained results by the enormous computer data up to $x=4\\times 10^{18}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.04940","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"}