⚠️ This page defines the mathematical objects used in this computational study. No proof of the Riemann Hypothesis is claimed. All results are computational observations requiring independent replication. ⚠️ Энэ хуудас тооцооллын судалгаанд ашигласан математикийн объектуудыг тодорхойлно. Риманы таамаглалын нотолгоо нэхэмжлэхгүй. Бүх үр дүн бие даасан давталт шаардсан тооцооллын ажиглалт юм.

Mathematical FormalismМатематикийн Формализм

Exact definitions, notation, and computational pipeline — v0.3 · May 2026Яг тодорхойлолт, тэмдэглэгээ, тооцооллын дараалал — v0.3 · 2026 оны 5-р сар

DefinitionsТодорхойлолтууд

Basic ObjectsУндсэн объектууд

Definition 1 — Riemann Zeros
Let \(\rho_n = \frac{1}{2} + i\gamma_n\) denote the non-trivial zeros of \(\zeta(s)\) with \(\gamma_n > 0\), ordered so that \(\gamma_1 \leq \gamma_2 \leq \cdots\)
Source: Odlyzko zero tables. Three blocks used: zeros_ht (T ≈ 74,920), zeros1 (T ~ 10¹²), zeros6 (T ~ 10¹³).
Definition 2 — Zero Spacings
\[\delta_n = \gamma_{n+1} - \gamma_n, \qquad \bar\delta = \frac{1}{N}\sum_{n=1}^{N}\delta_n\]
Raw spacings, not unfolded. Mean spacing varies with height T.
Definition 3 — Spacing Covariance at Lag h
\[C(h) = \frac{1}{N-h}\sum_{n=1}^{N-h}\bigl(\delta_n - \bar\delta\bigr)\bigl(\delta_{n+h} - \bar\delta\bigr)\]
The observable we compute. Evaluated at prime lags h = p.
Definition 4 — High-T Unfolded Normalization
\[\tau_p = \frac{\log p}{\log(T / 2\pi)}\]
Critical: using \(\tau_p = \log p / 2\pi\) (low-T formula) gives r ≈ 0.4–0.6. The high-T formula was selected after observing this — introducing potential selection bias.
Definition 5 — BK Predictor
\[B(p) = \frac{(\log p)^2}{p}\]
From Bogomolny–Keating (1996). Predicts the form of prime-dependent corrections to pair correlation statistics.
Definition 6 — Empirical Amplitude
\[A(p) = C(p)\]
The spacing covariance evaluated at prime lag p. We test whether A(p) ~ B(p).
ObservableАжиглагдах хэмжигдэхүүн

Main CorrelationУндсэн корреляц

\[r = \mathrm{Pearson}\bigl(\{A(p)\}_{p \leq 37},\; \{B(p)\}_{p \leq 37}\bigr)\]

Intermediate corrected values: r = 0.6847 (zeros1), r = 0.5113 (zeros_ht), r = 0.4509 (zeros6). Final result after Chebyshev ψ₀(x) validation: r = 0.017 (p = 0.95) — null result. Bootstrap MODERATE only at N=2M (zeros6); UNSTABLE at smaller N.

Statistical LimitationsСтатистикийн хязгаарлалтууд

Why These Results Are Not ConclusiveЯагаад эдгээр ур дүн дүгнэлт биш вэ

Limitation 1 — Selection BiasХязгаарлалт 1 — Сонголтын bias
The high-T normalization was chosen after observing that low-T gave weak results.
Limitation 2 — Small SampleХязгаарлалт 2 — Жижиг дээж
Pearson r with n = 12 data points is sensitive to outliers.
Limitation 3 — AutocorrelationХязгаарлалт 3 — Автокорреляц
Covariances C(p) and C(p′) for nearby primes are not independent. Effective DOF < 12.
Limitation 4 — Replication FailureХязгаарлалт 4 — Давталт амжилтгүй
Independent independent replication on Odlyzko 10¹² block gave r = 0.51, not r = 0.45–0.68 (dataset-dependent).
Computational PipelineТооцооллын дараалал

Step-by-StepАлхам алхмаар

01
Load zeros — Read γₙ from Odlyzko plain-text file.
02
Compute spacings — δₙ = γₙ₊₁ − γₙ, center: δₙ ← δₙ − δ̄.
03
Compute C(p) — For each prime p ≤ 37: C(p) = mean(δₙ · δₙ₊𝘑).
04
Compute B(p) — B(p) = (log p)² / p for each prime p.
05
Pearson r — Correlate {C(p)} and {B(p)} across 12 primes.
06
Null controls — Repeat with shuffled zeros, GUE surrogates, composite lags.
Explicit StatementТодорхой мэдэгдэл
Computational observations only — not a theorem, not a proof of RH. Independent independent replication and robustness testing are required.