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arXiv · 2601.16193

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

Gregorio Vettori
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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.

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Gregorio Vettori. 2026-01-22. Density-based structural frameworks for prime numbers, prime gaps, and Euler products. https://arxiv.org/abs/2601.16193

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