Hankel Determinants of Geometric Moments and Rational Lambert Values

Will Cook

Short paper

Hankel Determinants of Geometric Moments and Rational Lambert Values

Erdős #1049 By Will Cook September 2026 Authorship, AI use and citation

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Summary. Positive geometric moments and a uniform partition-sum bound give the fixed-base asymptotic of Zudilin’s normalised Hankel determinants, including its constant and power correction. A separate calculation of degrees after cancellation in his 2004 forms extends the rational-base irrationality region. It includes every positive integral power of $31 / 4$ and excludes $3 / 2$ .

This paper covers the fixed-base Hankel asymptotic and rational-base irrationality theorem; the long paper develops exact orders, content and endpoint obstructions, and the remaining problem at $3 / 2$ .

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

Introduction

Fix $q \in (0, 1)$ . Write $(z; q)_{m} = \prod_{j = 0}^{m - 1} (1 - z q^{j})$ and $F (t) = \sum_{n \geq 1} (t^{n} - 1)^{- 1}$ for $t > 1$ . At the auxiliary parameters $x = z = 1$ , the normalisation in [5] gives

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

The sums converge for $0 < q < 1$ . They also define formal series, since the $t$ th summand has $q$ -order $(m + 1) t$ .

Theorem 1.1 (the size of $V_{N}^{*}$ at a fixed base). Fix $0 < q < 1$ and write $P = (q; q)_{\infty}$ , $B_{N} = N (N - 1) (2 N - 1) / 6$ and $C_{N} = (N!)^{2} (N + 1)! / 2^{N}$ . There is a $K (q) > 0$ with

V_{N}^{*} (q) \sim K (q) C_{N} q^{B_{N}} P^{2 N} N^{- 8 F (1 / q)} (N \to \infty) .

The constant $K (q)$ is the convergent product in (13). Zudilin proved ${ord}_{q} V_{N}^{*} \geq B_{N}$ and the estimate $| V_{N}^{*} | \leq q^{N^{3} / 3} \exp (O_{q} (N^{2}))$ [5]. Theorem 1.1 determines the factorial, exponential and power factors contained in this estimate. We also recover the exact first formal term $C_{N} q^{B_{N}}$ . The formal calculation fixes $N$ and expands at $q = 0$ ; the asymptotic fixes $q$ and lets $N$ grow. Neither limit can be substituted for the other: multiplication by $(1 - q)^{N^{3}}$ preserves the first formal term but changes the fixed-base logarithm by a cubic quantity. No uniformity as $q \to 1$ is asserted.

The proof compares the moments with constant-weight geometric moments. We write $v_{m}^{*} = \sum_{k \geq 0} a_{k} q^{(m + 1) k}$ with $a_{k} > 0$ : the measure has mass $a_{k} q^{k}$ at $q^{k}$ . Heine’s identity expresses the determinant as a positive sum of squared Vandermondes over choices of $N$ nodes [7]. The first $N$ nodes normalise this sum but do not exhaust it; moving the last node gives a comparable contribution. Partitions encode all such displacements. For each fixed partition, only finitely many weight ratios change, at indices tending to infinity, so those ratios tend to $1$ . A summable bound independent of $N$ then makes the whole normalised sum asymptotically independent of the weights. Cauchy’s determinant formula evaluates its limit in the constant-weight case.

The weights retain a different contribution. Their factorisation involves two-fold and three-fold convolutions of $1 / (q; q)_{k}$ , with relative corrections $- 2 F (1 / q) / (k + 1)$ and $- 6 F (1 / q) / (k + 2)$ . In the logarithm of the weight product these give a harmonic sum, hence $N^{- 8 F (1 / q)}$ ; the summable errors affect only the constant. Theorem 2.1 separates the determinant argument from this calculation and also treats entries $(1 - q^{i + j + 1})^{- 2}$ .

Chowla’s conjecture, recorded by Erdős [1], asks whether $F (t)$ is irrational for every rational $t > 1$ . This is a related arithmetic question, and the fixed-base asymptotic alone does not answer it. In Section 4 we instead specialise the polynomial forms of Zudilin’s 2004 construction [4]. They prove irrationality for coprime $a > b \geq 1$ satisfying

\frac{\log b}{\log a} < θ^{*} = 0.4056830213840605 \dots,

where (15) defines $θ^{*}$ exactly. The constants are inherited from Zudilin; the degree calculation after polynomial cancellation supplies the rational-base estimate. The resulting region includes every positive integral power of $31 / 4$ , which lies outside the earlier sufficient region of Bundschuh and Väänänen [2]. Neither argument here settles $3 / 2$ .

Sections 2–3 prove Theorem 1.1; Section 4 gives the independent arithmetic argument. Section 5 links further questions to the companion. All logarithms are natural, empty products and determinants equal $1$ , and constants in $O_{q} (\cdot)$ may depend on the fixed $q$ .

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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