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

PDFOpen the PDF, 20 pages TeXLaTeX source Source on GitHub Problem page

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.

Start here, by the editorial route

In this paper

The weights of Zudilin’s moments

We apply Theorem 2.1 by collecting the dependence on $m$ in (1) into the single variable $w = q^{m + 1}$ . For this purpose, set

\begin{matrix} (11) & G_{q} (w) = \frac{1}{(w; q)_{\infty}^{3}} \sum_{t \geq 0} \frac{w^{t}}{(q; q)_{t}} \frac{(q^{t} w^{2}; q)_{\infty}}{(q^{t} w; q)_{\infty}^{2}}, a_{k} (q) = [w^{k}] P^{4} G_{q} (w) . \end{matrix}

The product identities $(q; q)_{m} = P / (q^{m + 1}; q)_{\infty}$ and $(q^{a}; q)_{m} = (q^{a}; q)_{\infty} / (q^{a + m}; q)_{\infty}$ give

\begin{matrix} (12) & v_{m}^{*} = P^{4} G_{q} (q^{m + 1}) = \sum_{k \geq 0} a_{k} (q) q^{(m + 1) k} . \end{matrix}

These identities hold formally in $Z [[q, w]]$ and, for fixed $0 < q < 1$ , analytically for $| w | < 1$ . In the formal identity only finitely many $t$ contribute to any $w$ -coefficient. For the analytic identity, we use the absence of denominator zeros in that disc and the locally uniform bounds on the product tails. The factor $w^{t}$ then gives locally uniform convergence of the sum. The signs of the coefficients are not evident from this product. The finite factorisation below gives positive weights and a uniform shift bound. Its two convolutions then give the accuracy needed to multiply the first $N$ weights.

Positive weights

The coefficient asymptotic

The first nonzero formal term

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.

Cite and contact. Cite this paper by its title, author and date above, with its PDF; cite the earlier sources it uses for a mathematical result. For software, use the release citation and give the commit used. Contact Will with questions or corrections.

Prefer the manuscript?

PDFOpen the PDF, 20 pages TeXOpen the LaTeX source Problem page

Equations typeset from the exact TeX. Rendered from the LaTeX manuscript source with file fingerprint sha256:7d9bf99ea6b7e699. Its file fingerprint matched the published source record, so the PDF above, the LaTeX, and this page are one manuscript. Redeploying the site regenerates this page from whatever the public repository holds at that moment.