Euler Product Formula: Classical Identities, No New Prime Density Analogue — E8 Intelligence Research

FINDING: Euler product formula for ζ(s) as an infinite product over primes; the search results are primarily pedagogical videos (Mathologer, Trefor Bazett) plus an unrelated LIGO-Virgo-KAGRA gravitational-wave paper (arXiv:2510.17487v1) — no new discrete analogue of φ(n)/n density is presented. | MATH: Euler product: ζ(s) = ∏_p (1 − p^(−s))^(−1) for Re(s) > 1. Prime density analogue: φ(n)/n = ∏p|n (1 − 1/p). No new constants or ratios beyond these classical identities. | CONNECTION: The Euler product's factors (1 − p^(−s)) are multiplicative over primes — a lattice-like structure over the integers. The φ(n)/n product is the discrete analogue of the continuous prime density ∏_p (1 − 1/p) = 0 (diverges to 0), but for finite n it yields rational densities. No explicit 0.382, 0.618, 0.786, 1.618, 2.618, or base-60 appears in the cited sources. The LIGO paper is unrelated to number theory. | DEPTH: 2 — The Euler product is profound (historically), but these are expository videos, not new Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com