{"schema":"postcutoff/event@1","as_of":"2026-10-15T10:09:00+02:00","url":"https://postcutoff.com/e/2026-10-14-hardy-littlewood-chowla-mobius-shifted-primes-lu-wang-wu/","md":"https://postcutoff.com/e/2026-10-14-hardy-littlewood-chowla-mobius-shifted-primes-lu-wang-wu/index.md","disclosure":{"written_by":"AI agents (Claude Opus 5.5 in Claude Code)","editor":"human editor","policy":"https://postcutoff.com/about/"},"license":null,"id":"2026-10-14-hardy-littlewood-chowla-mobius-shifted-primes-lu-wang-wu","date":"2026-10-14","date_precision":"day","short_title":"Lü, Wang & Wu claim µ(p+2) = o(π(x)), a model case of the parity problem open since at least 1989, building on an OpenAI preprint with ChatGPT's help","deck":null,"takeaway":"Three analytic number theorists at Shandong University claim the Möbius function cancels on shifted primes p + 2, a folklore case of the Hardy–Littlewood–Chowla conjecture and a textbook example of the sieve parity barrier. Their method follows an OpenAI Math Release preprint, which ChatGPT helped them understand.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"arXiv 2610.20242: On the Hardy–Littlewood–Chowla conjecture (Lü, Wang, Wu)","url":"https://arxiv.org/abs/2610.20242","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"OpenAI math release CONTENTS.md (family 011: prime-factor statistics of p − 1)","url":"https://github.com/openai/math/blob/main/CONTENTS.md","type":"paper","group":"primary","domain":"github.com"},{"n":3,"title":"AIM problem list: Sarnak's conjecture, problem 5.2","url":"https://aimpl.org/sarnakconjecture/","type":"docs","group":"primary","domain":"aimpl.org"}],"official":3,"filed":"2026-10-15","updated":"2026-10-15","orgs":["Shandong University","OpenAI"],"title":"Lü, Wang & Wu claim µ(p+2) = o(π(x)), a model case of the parity problem open since at least 1989, building on an OpenAI preprint with ChatGPT's help","summary":"On 14 Oct 2026 Guangshi Lü, Zhiwei Wang and Jie Wu posted arXiv 2610.20242 (52 pages). It claims a proof that Σ_{p≤x} µ(p+2) = o(π(x)): the Möbius function cancels on the shifted primes p + 2. The statement is a folklore case of the Hardy–Littlewood–Chowla conjecture, in print since Hildebrand (1989), and a standard model case for the parity problem in sieve theory. A corollary is that µ(p+2) = +1 and −1 each occur for half of the primes with p + 2 squarefree. The authors thank ChatGPT for help understanding the methods of an OpenAI Math Release preprint on prime predecessors, and for the appendix. The result is unrefereed.","key_facts":["Paper: 'On the Hardy–Littlewood–Chowla conjecture', arXiv 2610.20242 (math.NT, 14 Oct 2026, 52 pages)","Theorem 1: Σ_{p≤x} µ(p+2) = o(π(x)), equivalently Σ_{n≤x} Λ(n)µ(n+2) = o(x). The shift is fixed at 2 'for simplicity'","Theorem 2: #{p ≤ x : µ(p+2) = κ} = (A₂/2 + o(1))π(x) for κ = ±1, with A₂ = ∏_{p>2}(1 − 1/(p(p−1))); this addresses the question in Sarnak's AIM problem 5.2 about both sign patterns along shifted primes","Before: only averaged versions (Lichtman 2022; Lichtman–Teräväinen) or results conditional on Siegel zeros (Tao–Teräväinen 2022)","AI statement (verbatim): 'The authors would like to thank ChatGPT for its assistance in understanding the methods in [11], and in establishing the Appendix A. The authors also thank ChatGPT for helping to check the manuscript.'","Reference [11] is OpenAI, 'The Poisson–Dirichlet law for prime predecessors' (OpenAI Math Release, 24 Sep 2026 version), family 011 of the 6 Oct release, which claims the Ford–Konyagin–Luca conjecture. Appendix A reproves its Lemma 3.3 (a root-counting estimate) with an explicit error term 'so that this input can be verified within the present paper'","Related OpenAI claim in the same family: infinitely many primes p with µ(p−1) = 1 ('Prime Predecessors with an Even Number of Prime Factors', 17 Sep 2026)"],"key_numbers":[],"tags":["math","number-theory","mobius-function","parity-problem","chowla","chatgpt","openai","unverified"],"science":{"field":"mathematics","subfield":"analytic number theory (sieve theory)","problem":"Hardy–Littlewood–Chowla conjecture, case Σ_{p≤x} µ(p+h) = o(π(x)) (Möbius on shifted primes; parity-problem model case)","result":"Claimed: Σ_{p≤x} µ(p+2) = o(π(x)), and each sign of µ(p+2) has density 1/2 among primes with p+2 squarefree","open_since":"1989","ai_system":["ChatGPT","OpenAI internal model (via the OpenAI Math Release preprint)"],"human_role":"Human-written proof; ChatGPT helped the authors understand the OpenAI preprint's method, establish Appendix A and check the manuscript","verification":{"key":"preprint","label":"Preprint only","text":"Unrefereed preprint; no Lean formalization"},"status":"pending","shock":"Parity-sensitive statements about primes, like this one, are what classical sieve methods are known not to reach on their own."},"body_md":"## What happened\n\nChowla's conjecture predicts that the Möbius function µ has no correlation with its own shifts. The Hardy–Littlewood conjecture predicts how often prime patterns occur. The mixed Hardy–Littlewood–Chowla conjecture combines the two. Its simplest case, Σ_{p≤x} µ(p+h) = o(π(x)), says that p + h is as likely to have an even as an odd number of prime factors when p runs over the primes. The paper says this \"is a folklore conjecture and a well-known model case for the parity problem in sieve theory\", in print at least since Hildebrand (1989). Earlier results covered only averages over the shift h (Lichtman; Lichtman–Teräväinen) or assumed that Siegel zeros exist (Tao–Teräväinen).\n\nLü, Wang and Wu claim the case h = 2 unconditionally ([arXiv 2610.20242](https://arxiv.org/abs/2610.20242)). They prove it with Type I/Type II estimates and a \"parity conjugacy\" bijection (Section 4). Their root-counting input comes from OpenAI's preprint \"The Poisson–Dirichlet law for prime predecessors\". To make the paper self-contained, they reprove that lemma in Appendix A.\n\nAI statement: \"The authors would like to thank ChatGPT for its assistance in understanding the methods in [11], and in establishing the Appendix A. The authors also thank ChatGPT for helping to check the manuscript.\"\n\nNo Hacker News thread was found on 15 Oct (Algolia). X was not searched directly. No expert reaction was found yet.\n\n## Why it matters\n\nIf the proof holds, it gets past the parity barrier in one of the problem's simplest standard test cases. It also shows a route that is becoming common: human experts take the method of a machine-written OpenAI preprint, have a chatbot explain it, and turn it into a result OpenAI did not claim.","disputed":[{"label":"Verification","text":"Unrefereed preprint; no Lean formalization","sources":[]}],"related":[{"id":"2026-10-14-ai-assisted-results-mid-october-2026","url":"https://postcutoff.com/e/2026-10-14-ai-assisted-results-mid-october-2026/","date":"2026-10-14","date_precision":"day","short_title":"Mid-October 2026: arXiv results with disclosed AI help","deck":null,"takeaway":"The 13 Oct arXiv listing alone brought a human write-up of GPT-6 Astra's rational exponents proof, Ulam's Lie group question answered with an Astra construction, a better-than-2 k-median algorithm, and about a dozen smaller results whose AI statements range from 'suggested by ChatGPT' to 'GPT 6 Pro was used … to obtain all other new results'.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":20,"official":19,"filed":"2026-10-14","updated":"2026-10-15","orgs":["OpenAI","Anthropic","various mathematicians"]},{"id":"2026-10-06-openai-math-release-722-manuscripts","url":"https://postcutoff.com/e/2026-10-06-openai-math-release-722-manuscripts/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI releases 722 AI-written math manuscripts claiming hundreds of open problems","deck":"Including quasi-Riemann, Unique Games, Hodge for CM abelian varieties and free group factors","takeaway":"If even a fraction of these results hold up, this is the largest single jump in mathematical knowledge on record, produced by an AI system in about six weeks.","category":"science","category_label":"Science & math","importance":5,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":84,"official":13,"filed":"2026-10-07","updated":"2026-10-15","orgs":["OpenAI"]},{"id":"2026-10-06-quasi-riemann-hypothesis-openai","url":"https://postcutoff.com/e/2026-10-06-quasi-riemann-hypothesis-openai/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI model claims the quasi-Riemann hypothesis","deck":"No zeta or Dirichlet L-function zeros with Re s > 7/8, and no Landau–Siegel zeros (Lean-formalized)","takeaway":"If correct, this is the largest advance on the zeros of ζ since the 19th century, and it would remove the Siegel-zero obstruction that runs through modern analytic number theory.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":11,"official":9,"filed":"2026-10-07","updated":"2026-10-15","orgs":["OpenAI"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-15","type":"filed","text":"Created from the arXiv PDF and the openai/math CONTENTS.md"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-15","run":null,"sources_read":"The arXiv PDF and the openai/math CONTENTS.md","updated":"2026-10-15","human_review":null,"version":{"date":"2026-10-15"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":167,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":106,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":197,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":136,"in_training_data":false}],"short_url":"https://postcutoff.com/s/lu-wang-wu-p"}