Structural Obstructions in Fixed-Shift Prime Correlations via Mellin-Laplace Kernels
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911441933565952 |
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| author | Chen, Yung-Hua |
| author_facet | Chen, Yung-Hua |
| contents | This paper develops a Mellin-Laplace analytic framework for the fixed-shift prime correlation r_h(n) = Lambda(n) Lambda(n+h) for h not equal to 0. This sequence has no multiplicative structure, no Euler product, and no singularity at s = 1.
For every compactly supported Mellin-Laplace admissible kernel W, the smoothed shifted sum S_{W,h}(N) admits an absolutely convergent Mellin representation that holds entirely in the half-plane Re(s) > 1, with no use of analytic continuation.
The Mellin transform of W provides quantitative vertical decay, enabling full contour control on the boundary line Re(s) = 1 + eps. A Tauberian boundary analysis shows that both components of the boundary integral grow like N^{1+eps}, while the oscillatory part contributes an unavoidable N^{1+eps} (log N)^2 term. As a result, the boundary integral cannot be decomposed into a dominant main term plus a smaller error term, revealing a structural obstruction to main-term extraction for fixed-shift correlations.
These results give a complete analytic description of shifted prime correlations in their natural domain of convergence and clarify the analytic difficulties underlying problems such as the twin prime conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_00071 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Structural Obstructions in Fixed-Shift Prime Correlations via Mellin-Laplace Kernels Chen, Yung-Hua General Mathematics Primary 11N05, Secondary 11N13, 11N35, 11N37, 44A15 This paper develops a Mellin-Laplace analytic framework for the fixed-shift prime correlation r_h(n) = Lambda(n) Lambda(n+h) for h not equal to 0. This sequence has no multiplicative structure, no Euler product, and no singularity at s = 1. For every compactly supported Mellin-Laplace admissible kernel W, the smoothed shifted sum S_{W,h}(N) admits an absolutely convergent Mellin representation that holds entirely in the half-plane Re(s) > 1, with no use of analytic continuation. The Mellin transform of W provides quantitative vertical decay, enabling full contour control on the boundary line Re(s) = 1 + eps. A Tauberian boundary analysis shows that both components of the boundary integral grow like N^{1+eps}, while the oscillatory part contributes an unavoidable N^{1+eps} (log N)^2 term. As a result, the boundary integral cannot be decomposed into a dominant main term plus a smaller error term, revealing a structural obstruction to main-term extraction for fixed-shift correlations. These results give a complete analytic description of shifted prime correlations in their natural domain of convergence and clarify the analytic difficulties underlying problems such as the twin prime conjecture. |
| title | Structural Obstructions in Fixed-Shift Prime Correlations via Mellin-Laplace Kernels |
| topic | General Mathematics Primary 11N05, Secondary 11N13, 11N35, 11N37, 44A15 |
| url | https://arxiv.org/abs/2512.00071 |