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The research explores the crossover phenomenon in the performance of different mathematical representations of the Goldbach conjecture.

January 6, 20260 citationsOpen Access

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The Crossover Phenomenon in Hardy-Littlewood Goldbach Formula - Computational Evidence Across Five Orders of Magnitude

RCRuqing Chen

Key Points

The research explores the crossover phenomenon in the performance of different mathematical representations of the Goldbach conjecture.
Utilized a dataset of 21,511 verification points across five orders of magnitude from N = 10³ to N = 10⁸.
Employing ordered-pair counting methods and prime generation via the Sieve of Eratosthenes to 100 million primes.
Applied both series expansion and integral methods for computational comparisons.
Identified a critical crossover point at N ≈ 10⁵ for optimal method transition.
Demonstrated a peak accuracy advantage of 843-fold at N ≈ 5.5×10⁷.
Documented non-monotonic variation in advantage ratios, indicating complex arithmetic resonances.

Abstract

This dataset accompanies the research paper "The Crossover Phenomenon in Hardy-Littlewood Goldbach Formula: Computational Evidence of Scale-Dependent Performance and Asymptotic Dominance" by Ruqing Chen (2026). DATASET OVERVIEW: The dataset contains 21, 511 strategically sampled verification points spanning five orders of magnitude (N = 10³ to N = 10⁸), documenting a fundamental scale-dependent performance crossover between the classical Hardy-Littlewood series expansion and logarithmic integral formulation for predicting Goldbach representations. KEY FINDINGS DOCUMENTED: 1. Critical crossover phenomenon at N ≈ 10⁵ where optimal method transitions from series to integral2. Three distinct computational regimes with different error characteristics3. Peak accuracy advantage of 843-fold at favorable large-scale points (N ≈ 5. 5×10⁷) 4. Non-monotonic variation in advantage ratios (2-843×) revealing complex arithmetic resonances in asymptotic error structure5. Factor-of-2 correction in counting methodology (ordered pairs vs unordered) FILES INCLUDED: - COMPLETEDATASET₁kₜo₁00MFINAL. csv (2. 1 MB): Full dataset with 21, 511 points- THEFINALMASTERPIECE. png (6. 1 MB): Main 7-panel visualization (300 DPI) - THEFINALMASTERPIECEₕighres. png (17 MB): High-resolution version (600 DPI) - READMEFORZENODO. txt: Complete documentation METHODOLOGY: - Counting method: Ordered-pair Goldbach representations (consistent with Hardy-Littlewood circle method) - Prime generation: Sieve of Eratosthenes to 100M (5. 76M primes) - Series: 4th-order expansion in 1/log (N) - Integral: Adaptive Gaussian quadrature (limit=200) - Sampling: Stratified strategy across logarithmic scales- Validation: Cross-checked against OEIS A006307 DATA FORMAT (CSV columns): 1. N: Even integer tested2. SeriesBias: (Seriesₚrediction - Actual) / Actual3. IntegralBias: (Integralₚrediction - Actual) / Actual4. AbsBiasSeries: |SeriesBias|5. AbsBiasIntegral: |IntegralBias| USAGE: This dataset enables reproduction of all results in the paper and supports further research on Hardy-Littlewood asymptotic formulas, scale-dependent numerical methods, and arithmetic properties of Goldbach representations. CITATION: Chen, R. (2026). The Crossover Phenomenon in Hardy-Littlewood Goldbach Formula: Computational Evidence of Scale-Dependent Performance and Asymptotic Dominance. Journal to be added. Dataset: https: //doi. org/10. 5281/zenodo. 18123132 CONTACT: Ruqing ChenGUT Geoservice Inc. Montreal, Quebec, CanadaEmail: ruqing@hotmail. com CODE AVAILABILITY: Analysis code available at: https: //github. com/Ruqing1963/goldbach-crossover-phenomenon

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Ruqing Chen (Sat,) studied this question.

synapsesocial.com/papers/695d85413483e917927a43d9 https://doi.org/https://doi.org/10.5281/zenodo.18123131

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