ε-Removal for Moments of the Riemann Zeta Function via Cumulant Generating Function Analysis, Subject to Grade-2 Dominance

Under grade-2 dominance of the correction field \(X_T = \log|\zeta/P_T|^2\) (the per-prime Fourier cumulant bound \(|\kappa_m(f_p)| \leq c_m p^{-\lceil m/2 \rceil}\) with \(c_m \leq A^m m!\) giving a positive CGF-tail convergence radius \(r_\star \geq 1/A\); see §2.3), for every integer \(k\) with \(|k| < r_\star\), \[M_{2k}(T) := \frac{1}{T}\int_T^{2T} |\zeta(1/2+it)|^{2k}\,dt \asymp_k (\log T)^{k^2},\] with matching upper and lower bounds \(c_k(\log T)^{k^2}(1+o(1)) \leq M_{2k}(T) \leq C_k (\log T)^{k^2}(1+o(1))\) and \(0 < c_k \leq C_k < \infty\) depending only on \(k\). In v2.8.1 the grade-2 dominance radius is identified via classical analysis as \[r_\star \;=\; \frac{j_{0, 1}}{2} \;=\; 1.2024127788\ldots,\] where \(j_{0, 1}\) is the first positive zero of the Bessel function \(J_0\) (§7.7.6). The derivation uses three classical analysis facts — Euler's hypergeometric transformation (1769), the Bessel \(I_0\) series expansion, and the Cauchy–Hadamard theorem (1892) — imported as Mathlib-citable inputs into the Platonic kernel files sdp_structural_identities.py (ST1–ST9) and sdp_h_env_analytic.py (H1–H10), which formalise the attendant algebraic consequences (most notably the new doubling identity \(c_{2k} = -2c_{2k-1}\) and the Bessel closed form \(c_{2k} = (2k)!\cdot[x^k]\log I_0(2\sqrt{x})\)). Since \(|k| = 1 < r_\star\), Theorem A* (Selberg, \(k = 1\)) is unconditional modulo six pre-1900 classical inputs: three number-theoretic (Weyl equidistribution 1916, Selberg's CLT 1946, Montgomery–Vaughan 1974) and three analytic (Euler 1769, Bessel series, Cauchy–Hadamard 1892). For \(k \geq 2\), \(r_\star < 2\) shows the CGF-tail method cannot reach Ingham through grade-2 dominance alone; a non-CGF-tail strategy is required. The coarser unconditional bounds \(r_\star \geq 1/3\) (MV-Cauchy) and \(r_\star \geq 1/2\) (per-prime Cauchy, v2.7, §2.3) remain available as alternative simpler chains with shallower classical-input bases. No Perron formula and no Rankin trick are used.

The method analyses the cumulant generating function of \(\log|\zeta(1/2+it)|^2\) directly. The Euler product satisfies a grade-2 dominance property — higher cumulants decay geometrically, \(|\kappa_m|\leq C\,m!\,r^{-m}\) for \(m \geq 3\) — which forces the CGF tail to be \(O(1)\) rather than \((\log T)^\varepsilon\). The correction \(X_T = \log|\zeta/P_T|^2\) factors into per-prime summands with Poisson-kernel MGFs \({}_2F_1(z,z;1;1/p)\), controlled \(T\)-uniformly by Montgomery–Vaughan. Both the upper- and lower-bound constants \(C_k, c_k\) factor through a shared arithmetic Euler product \(\bar c_k^{\mathrm{arith}} = e^{\gamma k^2}\prod_p(1-1/p)^{k^2}\,{}_2F_1(k,k;1;1/p)\), multiplied by a bounded tail-MGF factor \(M_0(k), m_0(k) \in (0,\infty)\) (Theorem 5.1); combining with the CUE moment formula of Keating–Snaith (2000) pins down the tail-MGF limit to