English

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

Number Theory 2026-01-23 v1 Complex Variables

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.

Keywords

Cite

@article{arxiv.2601.16193,
  title  = {Density-based structural frameworks for prime numbers, prime gaps, and Euler products},
  author = {Gregorio Vettori},
  journal= {arXiv preprint arXiv:2601.16193},
  year   = {2026}
}

Comments

29 pages, 17 figures

R2 v1 2026-07-01T09:16:14.522Z