Informational Cardinality: A Unifying Framework for Set Theory, Fractal Geometry, and Analytic Number Theory
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866911716483268608 |
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| author | Li, Zhengqiang |
| author_facet | Li, Zhengqiang |
| contents | This paper investigates a class of deterministic fractals whose construction is governed by arithmetic sequences. We introduce the essential fractal prime set P_{ess} , a variant of the Cantor set constructed using the sequence of prime numbers modulo 4. We compute its Hausdorff dimension, \dim_H(P_{ess}) , and analyze its geometric complexity. In contrast to the classical middle-third Cantor set C_{1/3} , we demonstrate that while both sets are uncountable and share the same cardinality, their differing fractal dimensions (dim_H(C_{1/3}) versus the computed dimension of P_{ess}) reflect a fundamental difference in their geometric complexity. Furthermore, we propose a potential connection between the density of this prime-driven fractal and the distribution of zeros of the Riemann zeta function, formalized through the construction of a fractal zero set Z_F . This framework provides a novel geometric perspective on analytic number theory, illustrating how the fine-scale structure of primes can be encoded in deterministic fractal geometries. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_08587 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Informational Cardinality: A Unifying Framework for Set Theory, Fractal Geometry, and Analytic Number Theory Li, Zhengqiang General Mathematics 28A80, 11N05, 28A78 This paper investigates a class of deterministic fractals whose construction is governed by arithmetic sequences. We introduce the essential fractal prime set P_{ess} , a variant of the Cantor set constructed using the sequence of prime numbers modulo 4. We compute its Hausdorff dimension, \dim_H(P_{ess}) , and analyze its geometric complexity. In contrast to the classical middle-third Cantor set C_{1/3} , we demonstrate that while both sets are uncountable and share the same cardinality, their differing fractal dimensions (dim_H(C_{1/3}) versus the computed dimension of P_{ess}) reflect a fundamental difference in their geometric complexity. Furthermore, we propose a potential connection between the density of this prime-driven fractal and the distribution of zeros of the Riemann zeta function, formalized through the construction of a fractal zero set Z_F . This framework provides a novel geometric perspective on analytic number theory, illustrating how the fine-scale structure of primes can be encoded in deterministic fractal geometries. |
| title | Informational Cardinality: A Unifying Framework for Set Theory, Fractal Geometry, and Analytic Number Theory |
| topic | General Mathematics 28A80, 11N05, 28A78 |
| url | https://arxiv.org/abs/2603.08587 |