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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1410.1083","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-10-04T19:26:46Z","cross_cats_sorted":[],"title_canon_sha256":"c0b7d35efc68893a0ada41f35b8c2d374383a40521da5b5fa8cc4c2da600f8d2","abstract_canon_sha256":"2f4f1c7abec7f8940290a3bf785d8ffefa23a92d42ad9e581e6cd241f7f2e828"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:46:43.394171Z","signature_b64":"3TucoPfvhQvBMhBnECbqrk/QBRjqTk5uHGel+wEAHE35sgsUWYf817+BPbztWuOJBaF/POl+5q0LpivTGuhkDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e6aa9c652705babd330d267d3d3cb80ad6047f6648ef9e507b4c5147f74ba34e","last_reissued_at":"2026-05-18T00:46:43.393479Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:46:43.393479Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $\\mbox{li}(x)$. 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