Geometric Moments and Rational Lambert Values: Proofs and Further Results

Will Cook

Long paper

Geometric Moments and Rational Lambert Values: Proofs and Further Results

Erdős #1049 By Will Cook September 2026 Authorship, AI use and citation

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Summary. The record proves the fixed-base Hankel asymptotic for Zudilin's normalised moments and the rational-base irrationality criterion log b/log a<0.4056830213840605…. The determinant proof separates geometric node configurations from positive weights; the irrationality proof instead specialises Zudilin's 2004 polynomial forms after cancellation. Coefficient moments, evaluated lattices and congruence conditions distinguish integer divisibility from nonzero remainder decay. A broader ratio-limit argument and a rational-value model remain unformalised. None of these arguments settles $F (3 / 2)$ .

This paper covers the sharp fixed-base Hankel asymptotic and rational-base irrationality proofs, coefficient moments and arithmetic obstructions, with ordinary ratio-limit and rational-value models distinguished from checked results.

It does not establish the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or a solution to Erdős #1049, which remains open.

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Formal continued fractions and finite quadrature.

Write $D_{n} = D_{n, 0}$ , $E_{n} = D_{n, 1}$ , with $D_{0} = E_{0} = 1$ . All these determinant polynomials are nonzero by the all-rank degree calculation above. The classical Stieltjes continued fraction is thus defined over the rational-function field $Q (p)$ , with the convention

\sum_{m \geq 0} s_{m} (p) z^{m} = \frac{1}{1 - \frac{λ_{1} z}{1 - \frac{λ_{2} z}{1 - \dots}}},

λ_{2 n - 1} = \frac{E_{n} D_{n - 1}}{D_{n} E_{n - 1}}, λ_{2 n} = \frac{D_{n + 1} E_{n - 1}}{E_{n} D_{n}} (n \geq 1) .

Here $s_{0} = 1$ , and the identity is in $Q (p) [[z]]$ . The classical moment correspondence is recalled in [37]. Specialising a ratio at a fixed real $p$ requires its displayed denominators to be nonzero. The sixteen certificates ensure this and $λ_{1}, \dots, λ_{15} > 0$ for every $p \geq 1$ ; they do not ensure it at all indices. Seeking positive formulas for all these ratios is therefore a specific version of the coefficient-moment question. The generating series has radius of convergence zero for every $p \geq 1$ , so the displayed identity must be read formally. At $p = 1$ this follows from $s_{m} (1) = (m!)^{3}$ . For $p > 1$ , the $k = 0$ term of the positive expansion for $R_{m}$ and the inequality $[m]_{p}! \geq p^{m (m - 1) / 2}$ give

s_{m} (p) \geq p^{2 m^{2} - m} .

Thus $s_{m} (p)^{1 / m} \to \infty$ . This concerns the moment power series; it does not decide convergence of the continued fraction at an individual nonzero value of $z$ .

The same certificates also give a finite representing measure: for each fixed $p \geq 1$ , eight positive atoms reproduce exactly $s_{0}, \dots, s_{15}$ . The construction uses only the two certified $8 \times 8$ matrices, not positivity at higher ranks. Set

H_{0} = (s_{i + j})_{i, j < 8}, H_{1} = (s_{i + j + 1})_{i, j < 8}, T = H_{0}^{- 1} H_{1},

and give $R^{8}$ the inner product $⟨ u, v ⟩ = u^{T} H_{0} v$ . The operator $T$ is self-adjoint and positive definite because $H_{0} T = H_{1}$ is symmetric and positive definite. If $e_{0}, \dots, e_{7}$ are the coordinate vectors, the Hankel identities give $T e_{j} = e_{j + 1}$ for $0 \leq j < 7$ . Hence $e_{0}$ is cyclic: its first eight iterates form a basis. The spectral theorem now gives eight distinct positive eigenvalues and a positive weight at each, namely the squared norm of the corresponding projection of $e_{0}$ . Cyclicity ensures that no projection vanishes and that no eigenspace has dimension greater than one.

Let $ν$ be the resulting atomic measure. For $0 \leq m \leq 14$ , choose $i, j \leq 7$ with $i + j = m$ . Self-adjointness gives

\int x^{m} d ν (x) = ⟨ T^{m} e_{0}, e_{0} ⟩ = ⟨ e_{i}, e_{j} ⟩ = s_{m} .

For the last moment,

\int x^{15} d ν (x) = ⟨ T^{7} e_{0}, T T^{7} e_{0} ⟩ = e_{7}^{T} H_{1} e_{7} = s_{15} .

The weights sum to $s_{0} = 1$ . Hence the measure agrees with the moment functional on every polynomial of degree at most $15$ ; this is the degree range certified for the eight-node quadrature obtained here. In an orthonormal polynomial basis it is the classical Jacobi construction; Golub and Welsch describe the computation of its nodes and weights [41]. Nothing here identifies the subsequent moments of $ν$ with $s_{16}, s_{17}, \dots$ . This finite construction must also be distinguished from the coefficient pencil $(A_{N}, B_{N})$ . For rational modifications of an already known moment functional, see Krattenthaler [32]; such a modification producing the entire sequence $s_{m}$ has not been identified.

About this paper

Authorship and AI use. Will Cook built and directed the research infrastructure and maintains the public release. He reviewed claims when he could. AI agents did most of the research and drafting. Cook did not independently verify every claim.

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