The Quantum Geometry of Numbers

The Quantum Geometry of Numbers: Divisor Hilbert Spaces, Kernel Structures, and Entanglement Entropy of Arithmetic States Description: This work introduces a Hilbert-space framework for embedding natural numbers into a geometric and information-theoretic structure based on divisor relations. Each integer n is represented as a normalized quantum-inspired state over its divisors, inducing a natural geometry on N through inner products, kernels, and entropy measures. We define the divisor state|ₙ = 1d (n) ₃|₍ |d show that it generates a structured arithmetic geometry with the following properties: The kernel K (n, m) = d ( (n, m) ) is positive-definite and defines a reproducing kernel Hilbert space (RKHS) over the natural numbers. The induced Fubini–Study metric defines a nontrivial arithmetic distance on N. The exact bipartite von Neumann entropy satisfies S (n) = ₂ d (n). Highly composite numbers emerge as extremal high-entropy configurations in this geometric structure. This framework connects classical number theory (divisor functions, gcd structure), functional analysis (Hilbert spaces, RKHS theory), and quantum information concepts (state embeddings, entanglement entropy), without assuming any physical quantum system. The construction is fully deterministic and mathematically exact. Appendix A develops the formal Hilbert-space geometry and RKHS structure. Appendix B provides conceptual interpretations, analogies, and dynamical visualizations of divisor-state structure. Keywords: divisor function, number theory, Hilbert space embedding, reproducing kernel Hilbert space (RKHS), positive definite kernel, gcd kernel, arithmetic geometry, entropy, von Neumann entropy, quantum information theory, quantum-inspired mathematics, metric geometry on integers, highly composite numbers, spectral number theory.