The purpose of this work is to reveal a multi‑scale internal structure underlying the Riemann zeta function, a classical object of analytic number theory. Starting from the square ζ (x) ², we introduce a dyadic decomposition based on successive regroupments at scales 2, 4, 8, …, which naturally generates an infinite hierarchy of dyadic families of the form S (m, a, b, x) = Σ₊≥₁ (mk − a) ^-x (mk − b) ^-x. A key result is that each dyadic family admits a closed‑form expression in terms of polygamma functions and decomposes into a constant part and a finite linear combination of zeta values ζ (k+1) for 1 ≤ k ≤ x−1. Summing over all dyadic scales yields an exact quadratic identity ζ (x) ² = ζ (2x) + (22x+1/ (22x−1) ) · Z (x), where Z (x) collects all dyadic contributions. For even x, this identity involves only a single odd zeta value ζ (2x−1), with an explicit coefficient arising from the dyadic structure. This produces a family of explicit linear forms in odd zeta values, reminiscent of those appearing in the works of Rivoal, Ball–Rivoal, and Zudilin on the irrationality and linear independence of ζ (2n+1). Although no irrationality result is proved here, the dyadic decomposition provides a new structural framework in which such questions may be explored. The quadratic identity also leads to efficient numerical approximations of odd zeta values through truncated dyadic expansions. Overall, this work highlights a rich multi‑scale organisation of ζ (x) ² and suggests that dyadic methods may offer new perspectives on the longstanding mystery of odd zeta values.
Gael Ronsyn (Mon,) studied this question.