The Riemann hypothesis predicts that every nontrivial zero of the Riemann zeta function lies on the line with real part one half.
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Alpöge and Furman report an unconditional lower bound of 0.6725 for the proportion of nontrivial zeta zeros that are simple and lie on the critical line.
Original report: 2026-08-10OpenAI’s manuscript reports that all Dirichlet L-functions and finite-order Hecke L-functions over Q(√−3) have no zeros with real part greater than 7/8.
Read original source Original manuscript · family 003github.com
Original report: 2026-10-06The Riemann hypothesis predicts that every nontrivial zero of the Riemann zeta function lies on the line with real part one half.
The zeta function connects an analytic object with the prime numbers. The Riemann hypothesis says its nontrivial zeros have real part one half. The familiar convergent series defines zeta only in part of the complex plane; the full statement uses its analytic continuation.
The hypothesis constrains fluctuations in the distribution of primes. Checking many zeros provides finite evidence. A full proof must cover every nontrivial zero, and a counterexample must rigorously establish a zero away from the critical line. Conditional consequences should be clearly identified as conditional.
Its truth would sharpen our understanding of the distribution of prime numbers.
A valid proof, a counterexample, a substantial theorem, or a carefully scoped formalization. Checking finitely many zeros is not a proof of the full hypothesis.
Background review: 2026-10-11. Selected sources:
Publication dates and editorial changes.
Claude: a reported improvement in the proportion of zeta zeros on the critical line
Claude: a reported improvement in the proportion of zeta zeros on the critical line
Claude: a reported improvement in the proportion of zeta zeros on the critical line
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