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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An extension with limiting weight ratio

For $0 < ρ < q^{- 1}$ set

M (ρ; q) = \prod_{d \geq 1} (1 - ρ q^{d})^{- d}, M (1; q) = M (q) .

A limiting adjacent-weight ratio supplies an exponential shift bound, replacing the polynomial bound above. The condition $ρ q < 1$ makes that bound summable and changes one copy of the product constant from $M (1; q)$ to $M (ρ; q)$ . The ordinary proof below has no Lean proof in the supplied record; this unformalised remark is not used in the checked fixed-base argument.

Remark (limiting weight ratios; unformalised). Let $0 < q < 1$ and $0 < ρ < q^{- 1}$ . Suppose that $a_{k} > 0$ for $k \geq 0$ and

\frac{a_{k + 1}}{a_{k}} ⟶ ρ (k \to \infty) .

Define

M_{m} = \sum_{k \geq 0} a_{k} q^{(m + 1) k}, D_{N} = det (M_{i + j})_{0 \leq i, j < N} .

Then, as $N \to \infty$ ,

\begin{matrix} (21) & D_{N} \sim M (1; q)^{2} M (ρ; q) q^{B_{N}} P^{2 N} \prod_{k = 0}^{N - 1} a_{k} . \end{matrix}

Ordinary argument for the ratio-limit remark. Choose $\hat{ρ}$ with $ρ < \hat{ρ} < q^{- 1}$ . There is a $K$ such that $a_{j + 1} / a_{j} \leq \hat{ρ}$ for $j \geq K$ . With

C = \prod_{j < K} max (1, \frac{a_{j + 1}}{\hat{ρ} a_{j}}),

we obtain, by multiplying adjacent ratios,

\begin{matrix} (22) & \frac{a_{k + h}}{a_{k}} \leq C {\hat{ρ}}^{h} (k, h \geq 0) . \end{matrix}

This also gives $a_{h} \leq C a_{0} {\hat{ρ}}^{h}$ , so every moment converges because $q \hat{ρ} < 1$ . For finitely many nodes, Cauchy–Binet expands their moment determinant as a sum of squared Vandermonde products. We let the number of nodes tend to infinity. Entrywise convergence gives the limit of the determinant, while positivity gives the limit of the sum. Hence

\begin{matrix} (23) & D_{N} = \sum_{0 \leq k_{0} < \dots < k_{N - 1}} \prod_{i < N} a_{k_{i}} q^{k_{i}} \prod_{i < j < N} (q^{k_{i}} - q^{k_{j}})^{2} . \end{matrix}

The tuple $k_{i} = i$ contributes

q^{B_{N}} Δ_{N} \prod_{i < N} a_{i}, Δ_{N} = \prod_{d = 1}^{N - 1} (1 - q^{d})^{2 (N - d)} .

Write $k_{i} = i + λ_{i}$ and reverse the displacements: $μ_{j} = λ_{N - j}$ for $1 \leq j \leq N$ . Then $μ_{1} \geq \dots \geq μ_{N} \geq 0$ . After division by the least-tuple contribution, the summand becomes

\begin{matrix} (24) & \begin{aligned} W_{N} (μ) = & q^{\sum_{j \leq N} (2 j - 1) μ_{j}} \prod_{j \leq N} \frac{a_{N - j + μ_{j}}}{a_{N - j}} \\ \cdot \prod_{1 \leq j < k \leq N} {(\frac{1 - q^{k - j + μ_{j} - μ_{k}}}{1 - q^{k - j}})}^{2} . \end{aligned} \end{matrix}

For partitions of length greater than $N$ , set $W_{N} (μ) = 0$ . All the sums now have the same countable index set.

Suppose that $μ$ has $ℓ$ positive parts. Only the factors with $j \leq ℓ$ can differ from $1$ . For a fixed such $j$ , the denominators in the Vandermonde quotient multiply to at least $P$ , and its numerators are at most $1$ . We therefore have a bound independent of $N$

W_{N} (μ) \leq (C P^{- 2})^{ℓ} \prod_{j = 1}^{ℓ} {\hat{ρ}}^{μ_{j}} q^{(2 j - 1) μ_{j}} .

We sum this majorant by first fixing $ℓ$ and then dropping the ordering condition on the positive parts. This gives

\begin{aligned} \sum_{ℓ (μ) = ℓ} sup_{N} W_{N} (μ) & \leq (C P^{- 2})^{ℓ} \prod_{j = 1}^{ℓ} \sum_{u \geq 1} {\hat{ρ}}^{u} q^{(2 j - 1) u} \\ \leq C_{*}^{ℓ} q^{ℓ^{2}}, C_{*} = \frac{C \hat{ρ} P^{- 2}}{1 - \hat{ρ} q} . \end{aligned}

Here $\sum_{j \leq ℓ} (2 j - 1) = ℓ^{2}$ and $1 - \hat{ρ} q^{2 j - 1} \geq 1 - \hat{ρ} q > 0$ . The series $\sum_{ℓ \geq 0} C_{*}^{ℓ} q^{ℓ^{2}}$ converges.

For fixed $h$ , telescoping the adjacent ratios shows that $a_{k + h} / a_{k} \to ρ^{h}$ . Thus, for each fixed partition, the weight ratios in (24) have the same limit as for $a_{k} = ρ^{k}$ . The Vandermonde quotient is independent of the weights, and its convergence follows from the same finite identity (20). Dominated convergence therefore reduces the limit of $\sum_{μ} W_{N} (μ)$ to the case $a_{k} = ρ^{k}$ .

In that case $M_{m} = (1 - ρ q^{m + 1})^{- 1}$ , and Cauchy’s determinant gives

D_{N} = \frac{ρ^{N (N - 1) / 2} q^{B_{N}} Δ_{N}}{\prod_{0 \leq i, j < N} (1 - ρ q^{i + j + 1})} .

The multiplicity of $1 - ρ q^{d}$ in the denominator increases to $d$ , so the normalised sum tends to $M (ρ; q)$ . Finally,

\frac{Δ_{N}}{P^{2 N}} = \prod_{d < N} (1 - q^{d})^{- 2 d} \prod_{d \geq N} (1 - q^{d})^{- 2 N} ⟶ M (1; q)^{2},

since the logarithm of the second product is $O_{q} (N q^{N})$ . Combining the two limits proves (21). ◻

For example, $a_{k} = ρ^{k} (k + 1)^{s}$ , with $s \in R$ , satisfies the ratio hypothesis, and $\prod_{k < N} a_{k} = ρ^{N (N - 1) / 2} (N!)^{s}$ . For $s = 1$ the moments are $(1 - ρ q^{m + 1})^{- 2}$ , giving

det {((1 - ρ q^{i + j + 1})^{- 2})}_{i, j < N} \sim M (q)^{2} M (ρ; q) q^{B_{N}} P^{2 N} ρ^{N (N - 1) / 2} N! .

The choice $a_{k} = ρ^{k} \exp (\sqrt{k})$ also satisfies the ratio hypothesis, since $\sqrt{k + 1} - \sqrt{k} \to 0$ . For $ρ = 1$ , its relative growth at $k = 0$ exceeds every polynomial in the shift. Thus this argument extends beyond the polynomial-bound hypotheses of Theorem 3.6.

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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