← Session 3 — Project kickoff: RH blueprint, staged workflow + archive design
thinking (3037 chars)
Task 1.4 literature research: unconditional support-(1+δ) form-factor results. Per §4A.3, web research. Key question: is there ANY unconditional result giving Montgomery's pair-correlation form factor beyond Fourier support [−1,1], even partially?
Known landscape (from my knowledge):
- Montgomery 1973: form factor F(α) = ... for |α| ≤ 1 unconditional (F(α) = (1+o(1))T⁻²α log T + α for |α|≤1? Actually F(α) = α for 0≤α≤1 and the conjectured 1 for α ≥ 1).
- Under RH (or HL), known for α beyond 1.
- BGSTB24: "An unconditional Montgomery theorem for pair correlation" — they prove Montgomery's theorem (the F(α) for |α|≤1, sum over ALL complex zeros) unconditionally — support still ≤ 1.
- Goldston–Montgomery [GM87]: pair correlation conjecture ⟺ prime-pair (HL) information.
- What about support slightly beyond 1 with CURRENT unconditional technology? Known obstructions: the off-diagonal terms need cancellations among Λ(n)Λ(m) with n/m ≈ 1 up to X² — that's HL-type twin-prime correlations. No unconditional results exist; this is a famous wall.
- However: are there AVERAGED versions? E.g., "pair correlation in short intervals", or results by Goldston–Pintz–Yıldırım on prime gaps (they use second moments of primes with sieve weights — unconditional but that's about primes directly). GPY sieve: detects prime gaps via Selberg sieve second moments — unconditionally they proved small gaps (liminf (p_{n+1}−p_n)/log p_n = 0 — Zhang/Maynard bounded gaps). Do those give any support->1 form factor fragment? Maynard's sieve gives lower bounds for prime pairs in certain weighted averages... The connection: HL*(2) for X² ≤ T^{2−ε}... the additive correlation Σ_{n≤X} Λ(n)Λ(n+h) averaged over h up to H — unconditionally computable for H as large as...? There ARE unconditional results on AVERAGED prime correlations (e.g., Σ_{h≤H} Σ_n Λ(n)Λ(n+h) with H large is computable via circle method... no — singular series averages: Σ_{h≤H} 𝔖(h) ~ H is known unconditionally (the averaged singular series). The issue is the error terms per h.
Interesting angle: averaged pair correlations of primes. The form factor beyond support 1 involves Σ_{h} (Λ∗Λ−1)(h) f̂(h/H)... For the OFF-diagonal contribution in the second moment at X slightly > T: need Σ_{n≠m} Λ(n)Λ(m)/√(nm) w(n/m)... The obstruction is genuine — nobody has unconditional support > 1.
Also relevant: "Goldston–Suriajaya narrow box" GS26: zeros within o(1/log T) of line ⟹ results. And BGSTB25: "Pair correlation I: proportions of simple zeros and critical zeros".
Let me do the actual web research per protocol: arXiv API queries for recent papers on unconditional pair correlation beyond support 1. Use export.arxiv.org API.
Queries:
1. "unconditional pair correlation zeros" — recent
2. "form factor support greater than 1"
3. BGSTB25 arXiv:2501.14545, GLSS25 arXiv:2503.15449, GS25 arXiv:2511.20059, GS26 arXiv:2603.28104 — get abstracts.
Let me fetch the arXiv API for these specific IDs (they were cited in the paper) plus a search for newer unconditional results.
Starting **task 1.4 literature research** — the question: *does any unconditional support-(1+δ) form-factor fragment exist in the literature?* Per §4A.3, checking the primary sources cited by the paper plus searching for anything newer.
tool: WebFetch ok
input
{
"url": "http://export.arxiv.org/api/query?id_list=2501.14545,2503.15449,2511.20059&max_results=5",
"prompt": "For each paper: give the title, authors, date, and a faithful summary of the abstract. I need to know exactly what is proved UNCONDITIONALLY vs under hypotheses (RH, pair correlation conjecture, Hardy-Littlewood), and especially whether any result gives pair-correlation or form-factor information for the Riemann zeta function with Fourier support extending beyond [-1,1] unconditionally."
}
result 2026-08-17 21:48:16Z → 2026-08-18 01:26:18Z
## Paper 1 — arXiv:2501.14545v2 - **Title:** "Pair Correlation of Zeros of the Riemann Zeta Function I: Proportions of Simple Zeros and Critical Zeros" - **Authors:** Siegfred Alan C. Baluyot, Daniel Alan Goldston, Ade Irma Suriajaya, Caroline L. Turnage-Butterbaugh - **Date:** Published 2025-01-24; updated 2025-11-21 **Summary:** Montgomery's 1973 pair correlation theorem (proved under RH) gave that at least 2/3 of zeros are simple. This paper shows the pair correlation method can, for the first time, also address the "horizontal distribution" of zeros. **Nothing here is unconditional.** The results are proved under a hypothesis weaker than RH: that "all the zeros $ρ= β+iγ$ with $T<γ\le 2T$ are in a narrow vertical box centered on the critical line" of width $b/\log T$ with $b\to 0$. Under that condition they prove: - a generalization of Montgomery's "at least $2/3$ of zeros are simple"; - a new result that the method gives "at least $2/3$ of the zeros on the critical line"; - that "at least $1/3$ of the zeros are both simple and on the critical line" — which they note is "already known unconditionally using different methods." ## Paper 2 — arXiv:2503.15449v4 - **Title:** "Pair Correlation Conjecture for the Zeros of the Riemann Zeta-function I: Simple and Critical Zeros" - **Authors:** Daniel Alan Goldston, Junghun Lee, Jordan Schettler, Ade Irma Suriajaya - **Date:** Published 2025-03-19; updated 2026-03-30 **Summary:** Montgomery's Pair Correlation Conjecture (PCC) and his conjecture that ~100% of zeros are simple were originally justified using RH; Gallagher–Mueller (1978) showed PCC + RH implies 100% simplicity. This paper observes that "the method of Gallagher and Mueller does not depend on RH," so PCC alone unconditionally implies the simplicity conjecture. They treat PCC as a statement purely about vertical distribution, then show, via a new "horizontal multiplicity" idea, that PCC implies "asymptotically 100% of the zeros are not only simple but also on the critical line." **Everything is conditional on PCC**; no unconditional theorems are claimed. ## Paper 3 — arXiv:2511.20059v2 - **Title:** "Zeta Zeros on the Critical Line" - **Authors:** Daniel A. Goldston, Ade Irma Suriajaya - **Date:** Published 2025-11-25; updated 2026-02-05 (comment: "expository note, 9 pages") **Summary:** An expository note. It explains that "if RH could be removed from Montgomery's simple zero proof," the same argument would yield that "2/3 of the zeros are simple and on the critical line." It surveys how this idea underlies several recent papers. **No new unconditional results** — it is a conditional/heuristic overview. ## Your key question: Fourier support beyond [−1,1] unconditionally? **No.** None of these abstracts mention Fourier support ranges or form factors at all, and none claim any unconditional pair-correlation result for the zeta function. Every main result is conditional — on the narrow-box hypothesis (Paper 1), on PCC (Paper 2), or presented as a conditional remark (Paper 3). The only unconditional statement anywhere is the parenthetical note in Paper 1 that the 1/3 simple-and-critical proportion was previously known by other methods.
raw content_json (database row)
[{"type":"thinking","thinking":"Task 1.4 literature research: unconditional support-(1+δ) form-factor results. Per §4A.3, web research. Key question: is there ANY unconditional result giving Montgomery's pair-correlation form factor beyond Fourier support [−1,1], even partially?\n\nKnown landscape (from my knowledge):\n- Montgomery 1973: form factor F(α) = ... for |α| ≤ 1 unconditional (F(α) = (1+o(1))T⁻²α log T + α for |α|≤1? Actually F(α) = α for 0≤α≤1 and the conjectured 1 for α ≥ 1).\n- Under RH (or HL), known for α beyond 1.\n- BGSTB24: \"An unconditional Montgomery theorem for pair correlation\" — they prove Montgomery's theorem (the F(α) for |α|≤1, sum over ALL complex zeros) unconditionally — support still ≤ 1.\n- Goldston–Montgomery [GM87]: pair correlation conjecture ⟺ prime-pair (HL) information.\n- What about support slightly beyond 1 with CURRENT unconditional technology? Known obstructions: the off-diagonal terms need cancellations among Λ(n)Λ(m) with n/m ≈ 1 up to X² — that's HL-type twin-prime correlations. No unconditional results exist; this is a famous wall.\n- However: are there AVERAGED versions? E.g., \"pair correlation in short intervals\", or results by Goldston–Pintz–Yıldırım on prime gaps (they use second moments of primes with sieve weights — unconditional but that's about primes directly). GPY sieve: detects prime gaps via Selberg sieve second moments — unconditionally they proved small gaps (liminf (p_{n+1}−p_n)/log p_n = 0 — Zhang/Maynard bounded gaps). Do those give any support->1 form factor fragment? Maynard's sieve gives lower bounds for prime pairs in certain weighted averages... The connection: HL*(2) for X² ≤ T^{2−ε}... the additive correlation Σ_{n≤X} Λ(n)Λ(n+h) averaged over h up to H — unconditionally computable for H as large as...? There ARE unconditional results on AVERAGED prime correlations (e.g., Σ_{h≤H} Σ_n Λ(n)Λ(n+h) with H large is computable via circle method... no — singular series averages: Σ_{h≤H} 𝔖(h) ~ H is known unconditionally (the averaged singular series). The issue is the error terms per h.\n\nInteresting angle: averaged pair correlations of primes. The form factor beyond support 1 involves Σ_{h} (Λ∗Λ−1)(h) f̂(h/H)... For the OFF-diagonal contribution in the second moment at X slightly > T: need Σ_{n≠m} Λ(n)Λ(m)/√(nm) w(n/m)... The obstruction is genuine — nobody has unconditional support > 1.\n\nAlso relevant: \"Goldston–Suriajaya narrow box\" GS26: zeros within o(1/log T) of line ⟹ results. And BGSTB25: \"Pair correlation I: proportions of simple zeros and critical zeros\".\n\nLet me do the actual web research per protocol: arXiv API queries for recent papers on unconditional pair correlation beyond support 1. Use export.arxiv.org API.\n\nQueries:\n1. \"unconditional pair correlation zeros\" — recent\n2. \"form factor support greater than 1\"\n3. BGSTB25 arXiv:2501.14545, GLSS25 arXiv:2503.15449, GS25 arXiv:2511.20059, GS26 arXiv:2603.28104 — get abstracts.\n\nLet me fetch the arXiv API for these specific IDs (they were cited in the paper) plus a search for newer unconditional results.","signature":""},{"type":"text","text":"Starting **task 1.4 literature research** — the question: *does any unconditional support-(1+δ) form-factor fragment exist in the literature?* Per §4A.3, checking the primary sources cited by the paper plus searching for anything newer."},{"type":"tool_use","id":"WebFetch_224","name":"WebFetch","input":{"url":"http://export.arxiv.org/api/query?id_list=2501.14545,2503.15449,2511.20059&max_results=5","prompt":"For each paper: give the title, authors, date, and a faithful summary of the abstract. I need to know exactly what is proved UNCONDITIONALLY vs under hypotheses (RH, pair correlation conjecture, Hardy-Littlewood), and especially whether any result gives pair-correlation or form-factor information for the Riemann zeta function with Fourier support extending beyond [-1,1] unconditionally."},"extra_content":null}]