The Sharp Moment Hypothesis for the Riemann Zeta Function

This draft develops a global strategy aimed at the sharp upper bound \[\frac{1}{T}\int_T^{2T} |\zeta(1/2+it)|^{2k}\,dt \leq C(k)(\log T)^{k^2}\] for all integers \(k \geq 1\) and all \(T \geq T_0(k)\), where \(C(k)\) is an explicit constant depending only on \(k\). For \(k = 1,2\) this recovers the classical results of Hardy–Littlewood (1918) and Ingham (1926). For \(k \geq 3\), such a bound would sharpen Soundararajan's (2009) upper bound \(C_\varepsilon(\log T)^{k^2+\varepsilon}\) by removing the \(\varepsilon\).

The strategy works at the cumulant level. Write \(Y(t) = \log|\zeta(1/2+it)|\). A smoothed Selberg decomposition writes \(Y = Y_P + X\) where \(Y_P\) is a smoothed prime sum and \(X\) is a pointwise bounded remainder. The central analytic target is an Ibragimov-type bound \(T \cdot \alpha(T) \leq C_\alpha\) for an appropriate \(\alpha\)-coefficient between \(Y_P\) and \(X\), motivated by a cancellation mechanism: the Selberg variance \(\operatorname{Var}(Y_P) \sim \frac{1}{2}\log\log T\) appears in both a Cauchy–Schwarz covariance estimate and the Ibragimov normalization, so that the \(\sqrt{\log\log T}\) factors cancel in the algebra of Section 2. Granting this, a Billingsley-type cross-cumulant inequality controls the mixed terms in the Leonov–Shiryaev expansion and yields uniform bounds on \(|\kappa_m(Y)|\) for \(m \geq 3\), after which the cumulant generating function is dominated by its quadratic term and the \(k^2\) exponent follows upon evaluation at \(z = 2k\).

Apart from the author's assembly and the self-cited preprint below, the ingredients are standard results in the literature through Montgomery–Vaughan (2007). The novel contribution is the assembly: recognizing that the Selberg variance, an Ibragimov-type covariance bound, and a Billingsley-type cross-cumulant inequality may compose into a chain from the Euler product to the sharp moment upper bound. The algebraic chain recorded in the accompanying proof files (see frontmatter) is machine-verified (42 theorems, 21 facts, 0 errors); this does not, by itself, certify the analytic mixing step in Section 2.

Keywords: Riemann zeta function, moment hypothesis, cumulant generating function, Kronecker–Weyl equidistribution, additive–correlative duality

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