Prime Numbers as Causal Set Theory

Abstract Emergence. What is its origin? Where do organization and logarithmic patterns come from? Does it come from the four axioms and one conjecture of causal set theory, and are there other candidates that could describe the process of emergence? There are 10 quantum gravity models. The three spatial dimensions in Einstein’s equations — grr, gθθ, gφφ — already account for nine of those 10 models. LQG, GUP, DSR, Hagedorn, Asymptotic Safety, Hořava-Lifshitz, Non-commutative Geometry, CDT, and Standard GR — they’re all different proposed rules for how {x, y, z} behave at extreme curvature. Different answers to the same question. All living within the same three dimensions. The 10th — Causal Set Theory (CST) — stands apart. Rather than proposing new rules for spatial dimensions, CST is the underlying substrate from which GR itself emerges. In this paper we test whether an equation can satisfy CST’s four axioms and one conjecture. We test the Riemann Zeta function — ζ(s) = Σ 1/ns — and Euler’s product expression — ζ(s) = ∏ 1/(1 − p−s) for prime numbers. They satisfy all four axioms of CST, and they satisfy the conjecture.