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| Format: | Recurso digital |
| Sprog: | engelsk |
| Udgivet: |
Zenodo
2026
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| Fag: | |
| Online adgang: | https://doi.org/10.5281/zenodo.20143089 |
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- <p>v0.2.0 update (May 2026). This release derives the empirical finite-t correction of v0.1.0 from first principles. The Selberg constant B_at appearing in Var[S(γ_n)] = (log log(t/2π) + B_at)/(2π²) is now given a closed-form decomposition: B_at(∞) = B_continuous − π²/6 ≈ −0.2440, where B_continuous = 1 + γ + Σ_{m≥2} P(m)·(−1/m + 1/m²) ≈ +1.4010 is Goldston's (1987, Journal of Number Theory 27, 149–177) asymptotic constant for the continuous second moment of S(t). The geometric correction −π²/6 = −ζ(2) arises from the exact linearity of S(t) between consecutive zeros: in unfolded coordinates S has slope −1 across each unit interval, contributing ∫₀¹ (u − 1/2)² du = 1/12 per interval, or π²/6 in the (L + B)/(2π²) parametrization.</p> <p>The asymptotic value B_at(∞) = −0.2440 agrees with the empirical mean B_at = −0.2387 ± 0.002 measured across 8 datasets at heights 10⁶ ≤ t ≤ 10²², within O(1/log t) finite-t corrections. The empirical multiplier 0.956 from v0.1.0 is therefore not heuristic but precisely √((L + B_at_emp)/L) at typical heights.</p> <p>A new module riemann_bracket.constants exposes the analytical decomposition (prime zeta function, Goldston's prime sum, B_continuous, B_at_asymptotic) along with module-level cached values. The public API of v0.1.0 is unchanged; v0.1.0 users see no breaking changes.</p> <p>Empirical validation: across all 5 Platt bands and 3 Odlyzko datasets (8 total, 16 orders of magnitude in t), the ratio of empirical to theoretical standard deviation lies in [0.9989, 1.0016]. The single analytical constant B_at = −0.2387 calibrates all datasets to within 0.16% in standardised variance.</p> <p><br><br>A Python library that, given the index n of a non-trivial zero of the Riemann zeta function, returns an empirically-validated interval guaranteed to contain its t-coordinate. The predictor inverts the smooth zero-counting function targeting N(t) = n − 1/2, and σ_t is calibrated using Selberg's central limit theorem with an empirical finite-t correction factor of 0.956.</p> <p> </p> <p>The bracket has been validated across 16 orders of magnitude in t: 2,500,000 zeros at heights 10^6 to 10^10 (Platt) and 30,000 zeros at heights 10^12, 10^21, and 10^22 (Odlyzko). Across all four datasets, the maximum standardized residual |t_n − t_pred|/σ_t never exceeded 4.3, and a ±5σ window contains 100% of tested zeros without exception.</p> <p> </p> <p>Two structural observations emerge from this validation: (1) the residual distribution is sub-Gaussian at all tested heights, with empirical 4σ coverage exceeding the Gaussian prediction; (2) the empirical finite-t correction factor σ_emp/σ_Selberg increases monotonically from 0.953 to 0.969 across the tested range, consistent with convergence to 1 as predicted by Selberg's asymptotic theorem.</p> <p> </p> <p>The library is empirical: it provides no proven bound. The k=5 window is calibrated against observed data, not derived theoretically. The library does not compute zeros — it only brackets them; a separate zerofinder (Riemann-Siegel + Brent) is required to find the actual zero within the returned interval.</p>