Density-based structural frameworks for prime numbers, prime gaps, and Euler products
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866915747885744128 |
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| author | Vettori, Gregorio |
| author_facet | Vettori, Gregorio |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_16193 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Density-based structural frameworks for prime numbers, prime gaps, and Euler products Vettori, Gregorio Number Theory Complex Variables 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. |
| title | Density-based structural frameworks for prime numbers, prime gaps, and Euler products |
| topic | Number Theory Complex Variables |
| url | https://arxiv.org/abs/2601.16193 |