# Proportion of zeta zeros proved to lie on the critical line

A record target in qed.bot, the register of claims of AI work in mathematics. The register grades the evidence attached to each claim; it never rules on whether a proof is correct.

- Identifier: zeta:critical-line-proportion
- Collection: Analytic number theory
- Page: https://qed.bot/t/zeta-critical-line-proportion
- Record: 67.25%, held by Claude, with Levent Alpöge and Ralph Furman, since 10 Aug 2026
- Machine-checked: yes, by qed.bot
- Fidelity F2: The correspondence is declared through a Comparator challenge, an alignment table and written divergences.
- Source: https://arxiv.org/abs/2608.13637

## Record history

- 1974: more than 1/3, Norman Levinson; moved the record. https://doi.org/10.1016/0001-8708(74)90074-7
- 1989: more than 2/5, J. Brian Conrey; moved the record. https://doi.org/10.1515/crll.1989.399.1
- 22 Feb 2010: more than 41%, Hung Bui, Brian Conrey and Matthew Young; moved the record. https://arxiv.org/abs/1002.4127
- 28 Feb 2018: more than 5/12, Kyle Pratt, Nicolas Robles, Alexandru Zaharescu and Dirk Zeindler; moved the record. https://arxiv.org/abs/1802.10521
- 10 Aug 2026: 67.25%, Claude, with Levent Alpöge and Ralph Furman, with AI; moved the record. https://arxiv.org/abs/2608.13637

## AI claims

### Claude, 10 Aug 2026

Proved that at least two thirds of the zeta zeros are simple and on the critical line, 67.25% with a refined window, raising the proven proportion on the line from just over five twelfths; at least five sixths are distinct.

- Outcome: record
- Autonomy A3: Autonomous. The system produced the result without significant human mathematical involvement.
- Evidence V3: Machine-checked. The artifact was rebuilt at a pinned commit by someone other than its authors — qed.bot, or a registry that publishes the check — and it holds: a proof that elaborates on the standard axioms alone, or an object that meets the problem's own constraints. Checked by qed.bot.
- Fidelity F2: Declared. The authors record how the formal statement corresponds to the claim — a comparator challenge, an alignment table or written divergences — or a statement corpus cites the proof against its own statement.
- Source: Alpöge and Furman, More than two thirds of the zeta zeros are simple and on the critical line, https://arxiv.org/abs/2608.13637
- Source: Anthropic, Learning more about Claude's mathematical capabilities, https://www.anthropic.com/research/riemann-zeta
- Source: Lean formalisation, https://github.com/anthropics/formal-math/tree/main/zeta23

## Checks

- verified, by qed.bot: 23 theorems on the standard axioms only, at cec57f91 (https://github.com/anthropics/zeta-23-lean)

Cite as: qed.bot, "Proportion of zeta zeros proved to lie on the critical line", https://qed.bot/t/zeta-critical-line-proportion, as of 30 Sep 2026. The register's data is published under CC BY 4.0: https://creativecommons.org/licenses/by/4.0/