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

TL;DR

This paper introduces a Mellin-Laplace framework to analyze fixed-shift prime correlations, revealing structural obstructions that prevent extracting main terms, thus clarifying fundamental analytic difficulties in prime number conjectures.

Contribution

It develops a novel Mellin-Laplace analytic approach that fully describes fixed-shift prime correlations without relying on analytic continuation, exposing inherent structural obstructions.

Findings

Boundary integral grows like N^{1+eps}

Oscillatory component contributes N^{1+eps} (log N)^2

No decomposition into main term plus error is possible

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.