Heuristic Bounded Prime Gaps via a Chaotic Multidimensional Sieve and Random Matrix Theory

2025/07/23 by Milad Ghadimi, Ghadimi, Milad
Mathematics · Physics and Astronomy · #Advanced Mathematical Theories and Applications #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2507.17986

openalex publication_date 2025/07/23 · openalex created_date 2025/10/16 · openalex updated_date 2026/07/28

Abstract

We present the Enhanced Multidimensional Chaotic Heuristic Sieve (EMCHS), a novel probabilistic framework that integrates chaotic perturbations and random matrix theory (RMT) to suggest improved bounds on prime gaps. Building upon the foundational sieves of Goldston-Pintz-Yildirim and Maynard, EMCHS heuristically suggests unconditional gaps of at most 180 and conditional gaps of at most 8 under a partial Elliott-Halberstam conjecture (EHC) with delta = 0.3. These heuristic suggestions surpass Maynard's unconditional bound of 246 through refined polytope optimizations and probabilistic enhancements. We provide rigorous proofs for certain analytic components (such as bounding chaotic perturbations via ergodic theory) and explicitly distinguish which arguments and conclusions are heuristic or conjectural. Numerical evidence for primes up to 10¹8 supports the framework, and we discuss limitations and avenues for future rigorous work.

Citations

• Empirical verification of the even Goldbach conjecture and computation of prime gaps up to 4⋅10¹⁸
• Primes in Tuples I

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• Large prime gaps and probabilistic models
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