{"id": "card_n_639fcf317634", "kind": "note", "title": "Primes and zeta — Euler's bridge and the prime number theorem", "body": "Euler tied the primes to the continuum: zeta(s) = product over primes of 1/(1-p^-s), so a\nstatement about ALL integers becomes a statement about the primes. Sealed: there are exactly 25\nprimes below 100 and 168 below 1000 (primepi), and the prime number theorem's estimate\nx/ln(x) gives 144 for x=1000 — an ASYMPTOTIC undercount (true 168), honest about its own error:\nhttps://narrowhighway.com/s/51865a1cbf88345351a46cb24d1037fb156a2ea98bb0aa8723cde046de66aea9 . The primes thin out like 1/ln(x) but never stop (Euclid); the Riemann zeros control\nexactly how the actual count wobbles around the smooth estimate.", "source": {"label": "Concordance assay — 2026-07-09", "url": "https://narrowhighway.com/s/f8a7bbf4b8b3da9cadedda1d5228d6aa1683c60a1d79c3655c4eae419119276c", "ref": "riemann", "authority_tier": "engine_derived"}, "shelf": "science", "box": "riemann", "bands": ["prime number theorem", "euler product", "primepi", "primes", "zeta", "euclid"], "connections": [], "author": "engine", "created_at": "2026-07-10T01:49:27.180353+00:00", "updated_at": "2026-07-10T01:49:27.180353+00:00", "visibility": "public", "lifecycle_stage": "public", "volatility": "permanent", "surface": "secular", "metrics": {"paperclips_count": 0, "helpful_count": 0, "not_helpful_count": 0, "cite_count": 0, "walks_through_count": 0, "flagged_count": 0}, "source_hash": "d7f5836ef65a9a63d7161f853c0c4eb154b2b4ab10e556027243855c3dec308f"}