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A formally verified bound puts more than two thirds of zeta zeros on the critical line and simple

The manuscript proves unconditionally that at least two thirds of the nontrivial Riemann-zeta zeros, counted with multiplicity, are simple and on the critical line and that at least five sixths are distinct. An optimized window raises the constants to 0.6725 and 0.8362, and the results extend to primitive Dirichlet L-functions.

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Stable ID
math-zeta-two-thirds-simple-critical-line-2026
Revision
math-zeta-two-thirds-simple-critical-line-2026.v1
Field
Mathematics · Analytic number theory and formal verification
Evidence
Tier 1 · Peer reviewed: No
Record state
Current · Lean-verified preprint
Last checked

AI role

The paper states that Claude autonomously discovered and wrote the mathematical argument. Levent Alpöge and Ralph Furman independently studied and checked it, and a companion Lean 4 repository gives kernel-checkable statements and an audit trail.

Record details

Problem or result
Unconditional lower bounds for simple and distinct nontrivial zeros of the Riemann zeta function
Authors
Levent Alpöge and Ralph Furman; argument credited to Claude
Institutions
Anthropic hosted the project; the manuscript does not print formal author affiliations
Result date
Submitted August 13; manuscript dated August 17, 2026

Why it matters

The result raises the previous unconditional simple-on-line bound from 5/12 to above two thirds and supplies a formal artifact for a major analytic-number-theory advance without assuming the Riemann hypothesis.

Limits

This does not prove the Riemann hypothesis, simplicity of every zero, or any statement about the remaining fraction. It is a very recent preprint rather than a refereed article; formal verification establishes the encoded theorems, while specialists must still assess encoding, novelty and mathematical significance.

Sources

  1. Primary: Alpöge and Furman, zeta-zero lower-bound manuscript
  2. Formal proof: Anthropic Zeta23 Lean 4 formalization and audit

Correction and revision history

  1. 2026-08-22 — Added after primary-source, scope, status, AI-role and limitation review.

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