Riemann hypothesis

10 selected milestones · Source coverage checked 2026-10-10 · Historical annotation and mathematical review remain separate.

1. 1859 publication

   Foundation

   Riemann connects zeros to prime distribution

   The all-zero hypothesis arose alongside an explicit formula for counting primes.

   Scope & review: The conjectured all-zero location is not established by the explicit formula.

   Statements & sources

   • classical RH holds →

   • Bombieri · official RH description ↗§I, PDF page 1: Riemann’s 1859 memoir

   Historical annotation review: pending. The date refers to the publication, with the precision supported by the source.

2. Jan 18, 2018 preprint

   Bound established

   Newman’s nonnegativity conjecture

   Rodgers–Tao establishes Λ ≥ 0. The selected corrected paper preserves this theorem.

   Scope & review: The first submission date is historical; the cited v5 has a later correction. RH still requires Λ ≤ 0.

   Statements & sources

   • The Newman constant satisfies Λ ≥ 0 →

   • Rodgers–Tao · Newman constant is nonnegative ↗Theorem 1.1; §1 normalization and RH equivalence

   Historical annotation review: pending. The date refers to the preprint, with the precision supported by the source.

3. Apr 21, 2020 preprint

   Bound established

   Certified verification to height 3,000,175,332,800

   Platt–Trudgian uses rigorous arithmetic and counting to cover every zero through the specified ordinate; it also gives Λ ≤ 0.2.

   Scope & review: Height is not the number of checked zeros; no assertion about every higher zero follows.

   Statements & sources

   • RH verified to height 3,000,175,332,800 →
   • A cited upper bound Λ ≤ 0.2 →

   • Platt–Trudgian · rigorous finite verification ↗Theorem 1, p. 2; §2 interval/ball arithmetic and Turing counting
   • Platt–Trudgian · rigorous finite verification ↗Corollary 2, §3.4, p. 5; Polymath15 Table 1 applied to Theorem 1

   Historical annotation review: pending. The date refers to the preprint, with the precision supported by the source.

4. May 31, 2024 preprint

   Bound established

   Guth–Maynard zero-density estimate

   The first preprint introduced the improved exception-count estimate; the selected v2 and 2026 journal publication support the displayed statement.

   Scope & review: For fixed σ ≥ 3/4 this bounds possible off-line zeros; it does not exclude every exception.

   Statements & sources

   • Fewer possible zeros to the right of a fixed line →

   • Guth–Maynard · zero-density estimates ↗Theorem 1.2 and equation (1.4), p. 2

   Historical annotation review: pending. The date refers to the preprint, with the precision supported by the source.

5. Mar 23, 2026 preprint

   Bound established

   An explicit logarithmic zero-free region

   Bellotti–Trudgian–Yang excludes zeros near Re(s) = 1 with an explicit constant and t ≥ 3.

   Scope & review: The shrinking edge region does not cover the whole half-plane to the right of 1/2.

   Statements & sources

   • An explicit zero-free region near Re(s) = 1 →

   • Bellotti–Trudgian–Yang · explicit zero-free region ↗Theorem 1, p. 2 (strict inequality in theorem statement)

   Historical annotation review: pending. The date refers to the preprint, with the precision supported by the source.

6. Aug 19, 2026 preprint

   Bound established

   Above 67.25% simple zeros on the critical line

   Alpöge–Furman v2 presents the Claude-discovered argument, with human mathematical responsibility and reported Lean formalization.

   Scope & review: An asymptotic simple-and-on-line fraction over total multiplicity is not a percentage of RH solved. OpenMath has not executed the Lean code.

   Statements & sources

   • More than 67.25% simple zeros on the line, asymptotically →

   • Alpöge–Furman · simple zeros on the line ↗Theorem A, pp. 1–2; Montgomery–Taylor window; Remark 6.1
   • Alpöge–Furman · simple zeros on the line ↗Pinned Anthropic README, Solution.lean and formalization.yaml; Appendix A

   Historical annotation review: pending. The date refers to the preprint, with the precision supported by the source.

7. Sep 8, 2026 preprint

   Bound established

   A second proof and separate counting bounds

   Lamzouri v2 proves the same simple-on-line statement by a different argument and records further union and average bounds.

   Scope & review: A second proof does not count as a second increment of coverage. Simple OR on-line is not simple AND on-line.

   Statements & sources

   • More than 67.25% simple zeros on the line, asymptotically →
   • At least 88.762% simple OR on-line zeros, asymptotically →

   • Lamzouri · another proof and further proportions ↗Theorem 1.1; definitions of N, N₀ˢ in §1
   • Lamzouri · another proof and further proportions ↗Theorem 1.1; §1 definition of N_{s∪0}

   Historical annotation review: pending. The date refers to the preprint, with the precision supported by the source.

8. Oct 6, 2026 announcement

   Proposal

   OpenAI’s proposed seven-eighths half-plane

   The company release links a manuscript and formal evidence for a fixed zero-free half-plane Re(s) > 7/8.

   Scope & review: Independent acceptance and proof-chain correspondence are unassessed here. Even a valid bound would leave the region between 1/2 and 7/8.

   Statements & sources

   • Proposed result: a fixed zero-free half-plane Re(s) > 7/8 →

   • OpenAI · reported seven-eighths half-plane ↗Theorem 1.1; formalization scope family 003; comparator OAI.riemannZeta_ne_zero_of_seven_eighths_lt_re

   Historical annotation review: pending. The date refers to the announcement, with the precision supported by the source.

9. Date unavailable assessment

   Proposal

   An individual numerical Newman-bound candidate

   The pinned audit package proposes Λ ≤ 0.1787854 and attributes human–AI collaboration.

   Scope & review: The selected package has no independently established result date or accepted recompile here; the statement remains proposed.

   Statements & sources

   • Proposed result: Λ ≤ 0.1787854 →

   • Gomila · proposed improved Newman bound ↗README exact parameters; release manuscript; OPEN_REVIEW_QUESTIONS

   Historical annotation review: pending. The date refers to the assessment, with the precision supported by the source.

10. Date unavailable assessment

    Foundation

    From infinitely many zeros to positive proportions

    Bombieri’s historical account follows the Hardy, Selberg, Levinson and Conrey line of work.

    Scope & review: This is a grouped historical bridge with no single supported event date; the modern denominator and simplicity convention must still be read separately.

    Statements & sources

    • Bombieri · official RH description ↗§III: Hardy, Selberg, Levinson and Conrey; compare current AF Theorem A introduction

    Historical annotation review: pending. The date refers to the assessment, with the precision supported by the source.