analytic_number_theory

We reformulate the Moment Hypothesis for the Riemann zeta function as a cumulant boundedness condition and show that it reduces to a single distributional hypothesis about the Dirichlet polynomial approximation at $\sigma = 1/2$.

We show that the Moment Hypothesis for the Riemann zeta function — $m_{2k}(T) \leq C_k (\log T)^{k^2}$ for all $k \geq 1$ — follows from a single structural property of the cumulant generating function (CGF): its analyticity on a disk of radius $R > 0$ (equivalently, Latent diagnostic $\rho > 1$).