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

Complements and further questions

An unrestricted construction of small nonzero integer forms is another formulation of irrationality, not yet a method of proving it. The next problem remains equivalent to irrationality at $3 / 2$ even with its quadratic degree bounds, and even after imposing the exponential coefficient-height bound discussed below. We prove the equivalence before asking what a specified approximation family can supply.

Problem 10.1 (a divided linear form with small nonzero remainder). Exhibit an integer constant $C \geq 1$ and, for every sufficiently large positive integer $n$ , positive integers $W_{n}, R_{n}, S_{n}, M_{n}$ such that

n^{2} \leq W_{n}, R_{n}, S_{n} \leq C n^{2}, 4 R_{n} + 2 S_{n} \leq M_{n} \leq C n^{2},

together with polynomial pairs $(U_{n, j}, V_{n, j}) \in Z [X]^{2}$ for $0 \leq j < M_{n}$ , each of degree at most the common degree bound $W_{n}$ , whose specialised integer rows are primitive:

gcd (H_{W_{n}} (U_{n, j}), H_{W_{n}} (V_{n, j})) = 1.

Find a nonzero vector $λ^{(n)} \in {- 1, 0, 1}^{M_{n}}$ for which, on putting

U_{n} = \sum_{j < M_{n}} λ_{j}^{(n)} U_{n, j}, V_{n} = \sum_{j < M_{n}} λ_{j}^{(n)} V_{n, j},

the pair $(U_{n}, V_{n})$ is not $(0, 0)$ , all four residues vanish,

J_{3, R_{n}} (U_{n}) = J_{3, R_{n}} (V_{n}) = 0, J_{2, S_{n}} (U_{n}) = J_{2, S_{n}} (V_{n}) = 0,

where every residue in this display is formed using the common degree bound $W_{n}$ , and the resulting divided integer linear form

A_{n} = \frac{H_{W_{n}} (U_{n})}{3^{R_{n}} 2^{S_{n}}}, B_{n} = \frac{H_{W_{n}} (V_{n})}{3^{R_{n}} 2^{S_{n}}}, ρ_{n} = A_{n} F (3 / 2) - B_{n}

satisfies the explicit analytic condition

0 < | ρ_{n} | < \frac{1}{n} .

The residue equations make $A_{n}, B_{n}$ integers. A solution would prove irrationality: if $F (3 / 2) = a / b$ in lowest terms, every nonzero $ρ_{n}$ has absolute value at least $1 / b$ , contradicting the displayed bound for $n > b$ . Theorem 5.8 supplies only a nonzero signed relation with the four residue equations once the pairs and size inequality are present; it does not supply primitive input rows, a nonzero combined polynomial pair, or the nonvanishing and decay of $ρ_{n}$ .

Why the choice of family matters.

A congruence relation with nonzero remainder

Estimating the divided remainder

A restriction on Mahler functional equations

A limitation of the rectangular exponent model

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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