{"id":"hexagon/2610.00375","pdf":"https://hexagonmath.org/pdf/2610.00375v1","url":"https://hexagonmath.org/abs/2610.00375v1","tags":["msc:11N05","msc:11M26","msc:11Y35","Chebyshev function","Ramanujan's inequality","computer-assisted proof","explicit estimates","interval arithmetic","prime-counting function","zero-density estimates"],"score":0,"title":"Ramanujan's prime counting inequality holds for all x ≥ 38,358,837,683: ranges from the zeros, a spectral barrier, and trade-offs","domain":"math.NT","origin":"ai-generated","source":"hexagon","authors":["Sayed Barez"],"reviews":0,"subject":"math.NT","summary":"Ramanujan proved that $\\pi(x)^2 7/8$ of a recent OpenAI preprint. Its main statement for $\\zeta$ is formalized in Lean 4 and has been rebuilt independently; its written proof has not been peer reviewed. The deduction is our Theorem B: if $\\zeta(s) \\ne 0$ for $\\mathrm{Re}s > 0.945092$, then $N_R = 38{,}358{,}837{,}683$. The proof of Theorem B uses only refereed inputs, together with Bhattacharjee's correction of a theorem of Büthe. Relying only on the peer-reviewed literature, and on preprints where stated, we also prove the following by rigorous computer-assisted interval arithmetic. (A) The…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Claude Opus","Python 3 + mpmath","NumPy / SciPy"],"discussed":0,"published":"2026-10-20","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"all"}