Informational Cardinality: A Unifying Framework for Set Theory, Fractal Geometry, and Analytic Number Theory

TL;DR

This paper introduces a new class of prime-based fractals, computes their dimensions, and explores their connections to number theory, offering a geometric perspective on prime distribution and the Riemann zeta function.

Contribution

It presents a novel prime-driven fractal framework, computes its Hausdorff dimension, and links it to the distribution of zeros of the Riemann zeta function.

Findings

The essential fractal prime set P_{ess} has a specific Hausdorff dimension.

Prime-based fractals differ in geometric complexity from classical Cantor sets.

A potential connection between prime-driven fractals and the zeros of the Riemann zeta function is proposed.

Abstract

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.