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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In this paper

A positive expansion is not a moment representation.

Substituting $x = p^{m + 1}$ into Van Assche’s alternative expansion [18] gives the identity

R_{m} (p) = \sum_{k = 0}^{m} {[\binom{m}{k}]}_{p} {[\binom{m + k}{k}]}_{p} p^{(m - k) (m - k + 1) / 2} \prod_{j = m - k + 1}^{m} (p^{j} - 1) .

Indeed $(q x; q)_{k} = (- 1)^{k} \prod_{j = m - k + 1}^{m} (p^{j} - 1)$ , so the two signs cancel. This proves $R_{m} (p) > 0$ for $p > 1$ , and nonnegative coefficients in the variable $t = p - 1$ ; it does not prove Hankel positivity. In fact,

det (R_{i + j} (p))_{i, j < 3} = - 36 (p - 1)^{2} + O ((p - 1)^{3}) (p ↓ 1) .

A geometric factor $c ρ^{m}$ with $c, ρ > 0$ would change a Hankel matrix only by a positive scalar and an invertible diagonal congruence, so it would preserve positive definiteness. The factor $([m]_{p}!)^{3}$ is not geometric: its values at $m = 0, 1, 2$ are $1, 1, (1 + p)^{3}$ . It cannot be discarded in testing the moment condition. Nor does its positivity alone prove that multiplication by it repairs the failed condition for $R_{m}$ . At $p = 1$ , $s_{m} (1) = (m!)^{3}$ is the moment sequence of a product of three independent unit exponential variables, since their $m$ th moments multiply. This is only the polynomial endpoint; $F (1)$ diverges. Berg’s Theorem 5.1 states that $(m!)^{c}$ is Stieltjes indeterminate for $c > 2$ [36]. Thus a measure exists at the endpoint but is not unique. Neither uniqueness nor a canonical deformation is a premise of the question at $p > 1$ .

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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