Per-Prime Cumulant Structure of the Riemann Zeta Function

We study the cumulant structure of \(\log|\zeta(1/2+it)|^2\) induced by the Euler product. For each prime \(p\), define \(f_p(t) = 2\operatorname{Re}(-\log(1-p^{-1/2-it}))\). We prove three unconditional results: (1) the Kronecker--Weyl theorem gives exact cumulant additivity \(\kappa_m(\sum f_p) = \sum \kappa_m(f_p)\); (2) a reflection symmetry forces all cross-cumulants \(\kappa_{j,k}\) to vanish for odd \(k\); (3) \(\sum_p |\kappa_m(f_p)| < \infty\) for all \(m \geq 3\), placing the Euler product portion of \(\log|\zeta|^2\) in the mod-Gaussian convergence regime. The key idea is that the Fundamental Theorem of Arithmetic provides exact statistical independence via Kronecker--Weyl, while the per-prime bound \(|\kappa_m(f_p)| \leq m! \cdot 4^m p^{-m/2}\) and the convergence of \(\sum_p p^{-m/2}\) for \(m \geq 3\) do the rest. All 30 supporting theorems are machine-verified in a Python Lean 4 type checker with 0 novel axioms.

Keywords: Riemann zeta function, cumulants, Euler product, Kronecker--Weyl, mod-Gaussian convergence.

MSC 2020: 11M06 (primary), 60F05 (secondary).