The Composite Gamma Function: Analytical Framework, Critical Trench, and Prime Distribution Linkage
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| Natura: | Recurso digital |
| Lingua: | inglese |
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Zenodo
2025
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| _version_ | 1866901867980652544 |
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| author | Medford, Zach |
| author_facet | Medford, Zach |
| contents | <div dir="ltr"> <p>The paper introduces the <strong>Composite Gamma Function</strong>, Cf(z):=Γ(z+1)/eLi(z), which is an analytic continuation of the composite factorial into the complex plane. The central finding is the discovery of a <strong>"critical trench,"</strong> a vertical alignment of local minima consistently located along the line Re(z)=−40. This finding challenges predictions from standard asymptotic models, which incorrectly placed the trench at Re(z)=−41. The paper's core argument is that this <strong>1-unit discrepancy</strong> is resolved by incorporating the oscillatory correction terms from the Riemann-von Mangoldt explicit formula, which are governed by the non-trivial zeros of the Riemann Zeta function. The analysis establishes the Composite Gamma Function as an analytical bridge, making the subtle, oscillatory nature of prime number distribution directly manifest in the large-scale structure of composite numbers. The paper also explores the function's fundamental properties, including its meromorphic structure, pole analysis, and reflection symmetry. Additionally, it offers a dual perspective through the <strong>Composite Zeta Function</strong>, ζc(s), and shows that the values of this function at negative integers are encoded within the pole structure of Cf(z).</p> </div> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_16818383 |
| institution | Zenodo |
| language | eng |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | The Composite Gamma Function: Analytical Framework, Critical Trench, and Prime Distribution Linkage Medford, Zach Composite Gamma Function Analytic Number Theory Riemann Zeta Function Prime Distribution Complex Analysis Composite Numbers Critical Trench Gamma Function Logarithmic Integral <div dir="ltr"> <p>The paper introduces the <strong>Composite Gamma Function</strong>, Cf(z):=Γ(z+1)/eLi(z), which is an analytic continuation of the composite factorial into the complex plane. The central finding is the discovery of a <strong>"critical trench,"</strong> a vertical alignment of local minima consistently located along the line Re(z)=−40. This finding challenges predictions from standard asymptotic models, which incorrectly placed the trench at Re(z)=−41. The paper's core argument is that this <strong>1-unit discrepancy</strong> is resolved by incorporating the oscillatory correction terms from the Riemann-von Mangoldt explicit formula, which are governed by the non-trivial zeros of the Riemann Zeta function. The analysis establishes the Composite Gamma Function as an analytical bridge, making the subtle, oscillatory nature of prime number distribution directly manifest in the large-scale structure of composite numbers. The paper also explores the function's fundamental properties, including its meromorphic structure, pole analysis, and reflection symmetry. Additionally, it offers a dual perspective through the <strong>Composite Zeta Function</strong>, ζc(s), and shows that the values of this function at negative integers are encoded within the pole structure of Cf(z).</p> </div> |
| title | The Composite Gamma Function: Analytical Framework, Critical Trench, and Prime Distribution Linkage |
| topic | Composite Gamma Function Analytic Number Theory Riemann Zeta Function Prime Distribution Complex Analysis Composite Numbers Critical Trench Gamma Function Logarithmic Integral |
| url | https://doi.org/10.5281/zenodo.16818383 |