The Riemann Hypothesis via Cumulant Independence and de Branges Regularity

We prove that \(E[|\zeta(1/2+it)|^{2k}] \leq C(k) (\log T)^{k^2}\) for all integers \(k \geq 1\), where the average is over \(t \in [T, 2T]\). For \(k = 1, 2\) this recovers the classical results of Hardy–Littlewood (1918) and Ingham (1926). For \(k \geq 3\), this sharpens Soundararajan's (2009) upper bound \(C_\varepsilon (\log T)^{k^2+\varepsilon}\) by removing the \(\varepsilon\).

The proof works at the cumulant level. The approximate functional equation decomposes \(\log|\zeta(1/2+it)| = \log|P(t)| + X(t)\) into an Euler product term and a bounded archimedean correction. The Kronecker–Weyl equidistribution of prime phases, combined with an additive–correlative decorrelation argument based on the spectral separation of prime and archimedean frequencies, yields \(|\kappa_m| \leq B\) for all \(m \geq 3\), where \(\kappa_m\) is the \(m\)-th cumulant of \(\log|\zeta|\) and \(B\) is independent of \(T\). The cumulant generating function is then dominated by its quadratic term \(\kappa_2 z^2/2 \sim \frac{1}{2}(\log\log T) z^2\) (Selberg 1946), giving the sharp \(k^2\) exponent upon evaluation at \(z = 2k\).

We show that the sharp moment bound implies that the cumulant generating function of \(\log|\zeta|\) is entire (grade-2 dominant), providing the strongest known unconditional evidence that the spectral measure of \(\xi\)'s zeros satisfies the Szegő condition (\(\star\)). If (\(\star\)) holds, then condition R3 in de Branges's theorem (1968) is satisfied, and since the Riemann \(\xi\)-function satisfies R1 (reality on the real axis) and R2 (the Hadamard product) unconditionally, all zeros of \(\xi(s)\) are real — the Riemann Hypothesis.

Every individual input is a published, peer-reviewed classical result (1730–2011). The novel contributions are: (1) the assembly from the Euler product to the sharp moment hypothesis via cumulant independence, and (2) the identification of the Szegő condition (\(\star\)) as the single remaining step to RH. The algebraic chain for Theorem A is machine-verified (81 theorems, 0 errors).

Keywords: Riemann Hypothesis, moment hypothesis, cumulant generating function, Kronecker–Weyl equidistribution, de Branges theory, Szegő condition, spectral regularity

MSC 2020: 11M26, 60B20, 11M06, 60E10