{"paper":{"title":"Improving riemann prime counting","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Michel Planat (FEMTO-ST), Patrick Sol\\'e","submitted_at":"2014-10-04T19:26:46Z","abstract_excerpt":"Prime number theorem asserts that (at large $x$) the prime counting function $\\pi(x)$ is approximately the logarithmic integral $\\mbox{li}(x)$. In the intermediate range, Riemann prime counting function $\\mbox{Ri}^{(N)}(x)=\\sum_{n=1}^N \\frac{\\mu(n)}{n}\\mbox{Li}(x^{1/n})$ deviates from $\\pi(x)$ by the asymptotically vanishing sum $\\sum_{\\rho}\\mbox{Ri}(x^\\rho)$ depending on the critical zeros $\\rho$ of the Riemann zeta function $\\zeta(s)$. We find a fit $\\pi(x)\\approx \\mbox{Ri}^{(3)}[\\psi(x)]$ [with three to four new exact digits compared to $\\mbox{li}(x)$] by making use of the Von Mangoldt expl"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.1083","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":""},"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"}