Research log — Riemann hypothesis

Papers, leads, and failed lookups. arXiv is the record; GitHub repositories are implementation leads.

2026-08-30 — primary record

• Clay Mathematics Institute, Riemann hypothesis. Read the official problem page; CMI lists RH as unsolved.
• Rodgers–Tao, The De Bruijn-Newman constant is non-negative, arXiv:1801.05914v5. Fetched with scripts/arxiv_fetch.py. The paper proves $\Lambda\geq0$ and recalls that RH is equivalent to $\Lambda\leq0$.
• Polymath 15, Effective approximation of heat flow evolution of the Riemann $\xi$ function, and a new upper bound for the de Bruijn-Newman constant, arXiv:1904.12438v2. Fetched with the helper. Theorem 1.1 states $\Lambda\leq0.22$. Theorem 1.2 is the effective height/asymptotic-region/barrier criterion. Section 10, Table 1 prints the later $0.20$ parameter row.
• Platt–Trudgian, The Riemann hypothesis is true up to $3\cdot10^{12}$, arXiv:2004.09765v1. Fetched with the helper. Theorem 1 verifies RH through height $3{,}000{,}175{,}332{,}800$, covering the lowest $12{,}363{,}153{,}437{,}138$ zeros. Corollary 2 states $\Lambda\leq0.2$ using Polymath 15.

2026-08-30 — identifier corrections and failed lookups

• arXiv:2007.02194, 30 Years of Software Refactoring Research: A Systematic Literature Review. The supplied Platt–Trudgian identifier resolves to this unrelated computer-science paper. It was fetched before the mismatch was noticed and is not an RH source.
• arXiv:0809.1846, A removal lemma for systems of linear equations over finite fields. A guessed identifier for the Saouter–Gourdon–Demichel paper was unrelated. No arXiv version of that paper was located.

2026-08-30 — Lehmer-pair lower bound

• Saouter–Gourdon–Demichel, An improved lower bound for the de Bruijn-Newman constant, Math. Comp. 80 (2011), 2281–2287. Fetched the official AMS PDF. The paper proves the historical bound $\Lambda>-1.14541\cdot10^{-11}$ from a Lehmer pair near height $7.954\cdot10^{12}$. The four printed ordinates and Theorem A formula were replayed in compute/q1/. Rodgers–Tao's later $\Lambda\geq0$ dominates it.

2026-08-30 — implementation leads

• Jude Gomila, dbn-lambda-01787854-candidate-audit, commit a74738d. The repository proposes $\Lambda\leq0.1787854$ and explicitly labels the assembly as not yet peer reviewed. It is off arXiv, so this notebook treats it as a lead. The stored assembly passed; fresh tail, error, and barrier lanes were regenerated, and a fresh finite-producer run matched the first sealed row. The full finite range and analytic bridge were not independently replayed.
• K. M., dbn_upper_bound. Opened as an older implementation lead for the Polymath 15 computation. No number from this repository was promoted into the record.

2026-09-08 — independent certification campaign

• Reopened Polymath 15, arXiv:1904.12438v2 and fetched its PDF with the repository helper. Read Theorems 1.2 and 1.3, the error definitions, and the barrier approximation. PDF SHA-256: 60d0d2d381227d32f98535a5c43dedb86e333a9006d6819f0bdebdd734085603.
• Reopened Platt–Trudgian, arXiv:2004.09765v1 and fetched its PDF. Theorem 1 and Corollary 2 remain the zero-height input and published upper endpoint used here. PDF SHA-256: 3362f66af9fa9373977eee70e2282ec33989d5d8b97e0852df9e32cc25b52885.
• Reopened and cloned the candidate repository. Its current HEAD is still the q1 pin a74738deb6d5e0f76887cb36901da08b68dca705. Read the producer, proof notes, native binding, window and height transfers, tail lemma, and barrier source. Its use of the word proof is the author's claim, not independent acceptance.
• Searches for the exact candidate number with “arxiv” and for a 2026 de Bruijn–Newman upper bound did not locate an arXiv version of the candidate. Search results included the author's website, the existing notebook, and unrelated de Bruijn sequence papers. No search-result number was adopted as a new baseline.
• Completed the independent reproduction at the same pin. Both full finite algorithms and all fresh analytic numerical lanes passed. The source-level review also checks the barrier polynomial identity, geometric remainder denominators, and zero-time continuation. The result is recorded in compute/q2/CLAIM.md; this local verification does not change the candidate's arXiv or peer-review publication status.