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.

Start here, by the editorial route

In this paper

The sharp $q$ -order of the normalised Hankel determinant

We begin with the formal expansion of Zudilin’s normalised Hankel determinant at $x = z = 1$ . Determining its first nonzero term is a calculation in power series in $q$ ; an interpretation as a divisor after integer evaluation would additionally require an integral normalisation and a divisibility argument. Zudilin constructs the determinant by combining the Padé-type approximations of Bundschuh and Zudilin [12] with Bézivin’s method as developed in [13]; see [9]. His passage from the $q$ -order to the size of the determinant borrows the proofs of [13], as he notes in [9].

The source writes $ℓ_{p} (x, z) = x \sum_{r \geq 1} z^{r} / (p^{r} - x)$ , so $ℓ_{p} (1, 1) = F (p)$ . Here $p = q^{- 1}$ and $x, z$ are auxiliary parameters, not the base. Write $N$ for the Hankel rank. With $q$ a formal variable, the moments and determinant in question are

v_{m}^{*} = \sum_{t \geq 0} q^{(m + 1) t} \frac{(q; q)_{m}^{3} (q^{t + 1}; q)_{m}}{(q^{m + t + 1}; q)_{m + 1}}, V_{N}^{*} = det (v_{i + j}^{*})_{0 \leq i, j < N} .

All product denominators have constant term $1$ , so their inverses exist in $Z [[q]]$ . The order of a nonzero series is the least exponent of $q$ with a nonzero coefficient.

Theorem 3.4 (the first nonzero term at every rank). For every rank $N$ , the normalised Hankel determinant $V_{N}^{*}$ of [9], evaluated at $x = z = 1$ , has

{ord}_{q} V_{N}^{*} = \frac{N (N - 1) (2 N - 1)}{6}, leading coefficient \frac{(N!)^{2} (N + 1)!}{2^{N}} .

The source proves the order inequality and also gives $| V_{N}^{*} (q) | \leq | q |^{N^{3} / 3} \exp (O (N^{2}))$ , referring to additional estimates for the passage from formal order to analytic size [9]. Equality in the order bound rules out a further initial power of $q$ at $x = z = 1$ ; it does not itself prove a fixed- $q$ estimate. For $0 < q < 1$ , the positive-measure argument below gives both positivity and a two-sided comparison with the leading term, with logarithmic error $O_{q} (N)$ . It refines the subleading control, not the cubic exponent of $q$ .

The row identity.

Formal order and analytic size are different questions.

A separate positive-measure estimate.

Formal verification and computations.

Earlier work and the unresolved value at 3/23/2.

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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The sharp 𝑞q-order of the normalised Hankel determinant

About this paper

The sharp $q$ -order of the normalised Hankel determinant