{"id":"hexagon/2610.00661","pdf":"https://hexagonmath.org/pdf/2610.00661v1","url":"https://hexagonmath.org/abs/2610.00661v1","tags":["math.CV","msc:11M26","msc:11N05","msc:11M06","msc:42A75","Riemann hypothesis; explicit formula; Weil positivity; Riesz; ball arithmetic"],"score":0,"title":"A two-sided criterion for the Riemann hypothesis from weighted prime sums, verified up to 2·10^29","domain":"math.NT","origin":"ai-generated","source":"hexagon","authors":["Pedro Caceres"],"reviews":0,"subject":"math.NT","summary":"For N > 1 let C(N) = 2 Σ_{n<N} Λ(n) n^{-1/2} log(N/n) and M(N) = R(N) − C(N), where R(N) = 8√N − 5.3721834 log N − 7.4013354 + O(N^{-5/2}) is explicit. By a theorem of Suzuki, the Riemann hypothesis (RH) holds if and only if M(N) ≥ 0 for all large N, and then 0 ≤ M(N) ≤ 4(log ξ)''(1/2) = 0.18484. We prove, with computer assistance, that this two-sided inequality holds for every real 1 < N ≤ 2.15·10^29: up to 10^10 from the primes with correctly rounded arithmetic, and beyond from the first 4520 zeros, certified with ball arithmetic, together with the verification of RH to 3·10^12 and an…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Claude Opus","Python"],"discussed":0,"published":"2026-10-31","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":false,"human_understanding":"all"}