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Structural Obstructions in Fixed-Shift Prime Correlations via Mellin-Laplace Kernels

Yung-Hua Chen

TL;DR

The paper develops a Mellin-Laplace analytic framework to study fixed-shift prime correlations r_h(n)=Λ(n)Λ(n+h), demonstrating that the associated Dirichlet series R_h(s) lacks multiplicativity, Euler products, and a pole at s=1. By smoothing with a compactly supported kernel W and working in Re(s)>1, the authors obtain an absolutely convergent Mellin representation for the smoothed sum S_{W,h}(N) and reveal a structural boundary obstruction: the boundary integral on Re(s)=1+ε cannot be decomposed into a dominant main term plus a smaller error term, with the oscillatory part carrying an unavoidable N^{1+ε}log^{2}N growth. This obstruction is intrinsic to the analytic domain and not a technical artifact of kernel choice, explaining why standard contour methods fail to yield a main term for fixed-shift correlations like twin primes. The framework isolates the analytic ingredients available in Re(s)>1 and clarifies what additional arithmetic, spectral, or probabilistic input would be required to overcome the barrier in future work.

Abstract

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.

Structural Obstructions in Fixed-Shift Prime Correlations via Mellin-Laplace Kernels

TL;DR

The paper develops a Mellin-Laplace analytic framework to study fixed-shift prime correlations r_h(n)=Λ(n)Λ(n+h), demonstrating that the associated Dirichlet series R_h(s) lacks multiplicativity, Euler products, and a pole at s=1. By smoothing with a compactly supported kernel W and working in Re(s)>1, the authors obtain an absolutely convergent Mellin representation for the smoothed sum S_{W,h}(N) and reveal a structural boundary obstruction: the boundary integral on Re(s)=1+ε cannot be decomposed into a dominant main term plus a smaller error term, with the oscillatory part carrying an unavoidable N^{1+ε}log^{2}N growth. This obstruction is intrinsic to the analytic domain and not a technical artifact of kernel choice, explaining why standard contour methods fail to yield a main term for fixed-shift correlations like twin primes. The framework isolates the analytic ingredients available in Re(s)>1 and clarifies what additional arithmetic, spectral, or probabilistic input would be required to overcome the barrier in future work.

Abstract

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.

Paper Structure

This paper contains 12 sections, 3 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Historical background
  3. Structural obstruction
  4. Contribution of this work
  5. The Mellin--Laplace Analytic Framework
  6. The Mellin--Laplace admissible kernel family
  7. Mellin transform and vertical decay
  8. The main analytic identity
  9. Structural Obstruction for Shifted Correlations
  10. Analytic summary.
  11. Conceptual interpretation.
  12. Conclusion and Outlook

Key Result

Lemma 2.3

For a Mellin--Laplace kernel $W$, the transform $M_W(s)$ is holomorphic on the half–plane $\Re s>0$ and satisfies, for each fixed $\sigma>0$, with implied constants depending only on $\sigma$ and $\|\psi\|_{C^{2}}$.

Theorems & Definitions (11)

  • Definition 2.1: Mellin--Laplace admissible kernel family
  • Remark 2.2: Natural analytic origin of the kernel
  • Lemma 2.3: Vertical decay
  • proof
  • Theorem 2.4: Mellin--Laplace identity
  • proof
  • Remark 2.5: Analytic interpretation
  • Theorem 3.1: Intrinsic limitations on the boundary decomposition
  • proof
  • Remark 3.2: Kernel-independence of the obstruction
  • ...and 1 more