The Crossover Phenomenon in Hardy-Littlewood Goldbach Formula - Computational Evidence Across Five Orders of Magnitude
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| Formato: | Recurso digital |
| Lenguaje: | inglés |
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2026
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| _version_ | 1866901340046753792 |
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| author | Chen, Ruqing |
| author_facet | Chen, Ruqing |
| contents | <p>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).</p> <p>DATASET OVERVIEW:<br>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.</p> <p>KEY FINDINGS DOCUMENTED:<br>1. Critical crossover phenomenon at N ≈ 10⁵ where optimal method transitions from series to integral<br>2. Three distinct computational regimes with different error characteristics<br>3. Peak accuracy advantage of 843-fold at favorable large-scale points (N ≈ 5.5×10⁷)<br>4. Non-monotonic variation in advantage ratios (2-843×) revealing complex arithmetic resonances in asymptotic error structure<br>5. Factor-of-2 correction in counting methodology (ordered pairs vs unordered)</p> <p>FILES INCLUDED:<br>- COMPLETE_DATASET_1k_to_100M_FINAL.csv (2.1 MB): Full dataset with 21,511 points<br>- THE_FINAL_MASTERPIECE.png (6.1 MB): Main 7-panel visualization (300 DPI)<br>- THE_FINAL_MASTERPIECE_highres.png (17 MB): High-resolution version (600 DPI)<br>- README_FOR_ZENODO.txt: Complete documentation</p> <p>METHODOLOGY:<br>- Counting method: Ordered-pair Goldbach representations (consistent with Hardy-Littlewood circle method)<br>- Prime generation: Sieve of Eratosthenes to 100M (5.76M primes)<br>- Series: 4th-order expansion in 1/log(N)<br>- Integral: Adaptive Gaussian quadrature (limit=200)<br>- Sampling: Stratified strategy across logarithmic scales<br>- Validation: Cross-checked against OEIS A006307</p> <p>DATA FORMAT (CSV columns):<br>1. N: Even integer tested<br>2. Series_Bias: (Series_prediction - Actual) / Actual<br>3. Integral_Bias: (Integral_prediction - Actual) / Actual<br>4. Abs_Bias_Series: |Series_Bias|<br>5. Abs_Bias_Integral: |Integral_Bias|</p> <p>USAGE:<br>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.</p> <p>CITATION:<br>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</p> <p>CONTACT:<br>Ruqing Chen<br>GUT Geoservice Inc.<br>Montreal, Quebec, Canada<br>Email: ruqing@hotmail.com</p> <p>CODE AVAILABILITY:<br>Analysis code available at: https://github.com/Ruqing1963/goldbach-crossover-phenomenon</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18123132 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | The Crossover Phenomenon in Hardy-Littlewood Goldbach Formula - Computational Evidence Across Five Orders of Magnitude Chen, Ruqing Goldbach conjecture Hardy-Littlewood formula computational number theory prime number distribution asymptotic analysis crossover phenomenon series expansion logarithmic integral numerical verification additive number theory Number theory Computational science <p>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).</p> <p>DATASET OVERVIEW:<br>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.</p> <p>KEY FINDINGS DOCUMENTED:<br>1. Critical crossover phenomenon at N ≈ 10⁵ where optimal method transitions from series to integral<br>2. Three distinct computational regimes with different error characteristics<br>3. Peak accuracy advantage of 843-fold at favorable large-scale points (N ≈ 5.5×10⁷)<br>4. Non-monotonic variation in advantage ratios (2-843×) revealing complex arithmetic resonances in asymptotic error structure<br>5. Factor-of-2 correction in counting methodology (ordered pairs vs unordered)</p> <p>FILES INCLUDED:<br>- COMPLETE_DATASET_1k_to_100M_FINAL.csv (2.1 MB): Full dataset with 21,511 points<br>- THE_FINAL_MASTERPIECE.png (6.1 MB): Main 7-panel visualization (300 DPI)<br>- THE_FINAL_MASTERPIECE_highres.png (17 MB): High-resolution version (600 DPI)<br>- README_FOR_ZENODO.txt: Complete documentation</p> <p>METHODOLOGY:<br>- Counting method: Ordered-pair Goldbach representations (consistent with Hardy-Littlewood circle method)<br>- Prime generation: Sieve of Eratosthenes to 100M (5.76M primes)<br>- Series: 4th-order expansion in 1/log(N)<br>- Integral: Adaptive Gaussian quadrature (limit=200)<br>- Sampling: Stratified strategy across logarithmic scales<br>- Validation: Cross-checked against OEIS A006307</p> <p>DATA FORMAT (CSV columns):<br>1. N: Even integer tested<br>2. Series_Bias: (Series_prediction - Actual) / Actual<br>3. Integral_Bias: (Integral_prediction - Actual) / Actual<br>4. Abs_Bias_Series: |Series_Bias|<br>5. Abs_Bias_Integral: |Integral_Bias|</p> <p>USAGE:<br>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.</p> <p>CITATION:<br>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</p> <p>CONTACT:<br>Ruqing Chen<br>GUT Geoservice Inc.<br>Montreal, Quebec, Canada<br>Email: ruqing@hotmail.com</p> <p>CODE AVAILABILITY:<br>Analysis code available at: https://github.com/Ruqing1963/goldbach-crossover-phenomenon</p> |
| title | The Crossover Phenomenon in Hardy-Littlewood Goldbach Formula - Computational Evidence Across Five Orders of Magnitude |
| topic | Goldbach conjecture Hardy-Littlewood formula computational number theory prime number distribution asymptotic analysis crossover phenomenon series expansion logarithmic integral numerical verification additive number theory Number theory Computational science |
| url | https://doi.org/10.5281/zenodo.18123132 |