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

Coefficient moments and cyclotomic content

We now study the coefficient sequence, not the remainder sequence. The identity $v_{m}^{*} = α_{m} F (p) - β_{m}$ does not transfer a representing measure from $(v_{m}^{*})$ to $(α_{m})$ : subtraction mixes two coefficient sequences. Their positivity, spectral properties and cyclotomic factors therefore require separate calculations. Fix $p > 1$ and define, using Gaussian binomial polynomials,

\begin{aligned} R_{m} (p) & = \sum_{k = 0}^{m} (- 1)^{m + k} p^{k (k + 1) / 2} {[\binom{m}{k}]}_{p} {[\binom{m + k}{k}]}_{p}, \\ s_{m} (p) & = ([m]_{p}!)^{3} R_{m} (p), \end{aligned}

where $[m]_{p}! = \prod_{j = 1}^{m} (1 + p + \dots + p^{j - 1})$ . Each $R_{m}$ is the signed value at $x = p^{m + 1}$ of the little $q$ -Legendre polynomial of degree $m$ , with $q = p^{- 1}$ ; this identification is discussed in the related-work section. The polynomial normalisation of [9], at $x = z = 1$ , gives

\begin{aligned} α_{m} & = p [p (p - 1)^{3}]^{m} s_{m} (p), \\ β_{m} & = {[α_{m} \sum_{j \geq 1} τ (j) p^{- j}]}_{+} - 1, \\ v_{m}^{*} & = α_{m} F (p) - β_{m} . \end{aligned}

The brackets mean the polynomial part at infinity. Thus $α_{0} = p$ and $β_{0} = 0$ . This fixes the normalisation before any content or positivity test.

A positive expansion is not a moment representation.

Finite certificates and an all-rank degree check.

A finite spectral consequence.

Formal continued fractions and finite quadrature.

Testing for additional cyclotomic factors.

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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