Density-based structural frameworks for prime numbers, prime gaps, and Euler products

TL;DR

This paper introduces a unified density-based framework for understanding prime numbers, prime gaps, and Euler products, revealing structural tensions and providing bounds and interpretations aligned with classical conjectures and the Riemann zeta function.

Contribution

It develops a novel density-based model connecting prime distribution, prime gaps, and Euler products, offering new bounds and insights into prime number conjectures.

Findings

Quantitative estimates on the rarity of extreme prime gaps.

Bounds on additive representations of even integers consistent with Hardy-Littlewood heuristics.

A new interpretation of the Riemann zeta function's critical line via truncated Euler products.

Abstract

We develop a unified density-based framework for primality, coprimality, and prime pairs, and introduce an intrinsic normalized model for prime gaps constrained by the Prime Number Theorem. Within this setting, a structural tension between Hardy-Littlewood, Cramer, and PNT predictions emerges, leading to quantitative estimates on the rarity of extreme gaps. Additive representations of even integers are reformulated as local density problems, yielding non-conjectural upper and lower bounds compatible with Hardy-Littlewood heuristics. Finally, the Riemann zeta function is analyzed via truncated Euler products, whose stability and oscillatory structure provide a coherent interpretation of the critical line and prime-based numerical criteria for the localization of non-trivial zeros.