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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Further questions

The determinant asymptotic still needs a coefficient-denominator estimate to give an integer contradiction. The companion’s rational-value model has the same formal orders, leading coefficients and type of fixed-base power correction, but a rational target value. Its denominator calculation shows why clearing need not yield small nonzero integer forms. The model and its denominator argument have ordinary proofs only and are not used above.

The approximants’ polynomial coefficients pose a separate moment problem. The normalisation matters in Van Assche’s little- $q$ -Legendre construction [12] and the multiple-orthogonality construction of Postelmans and Van Assche [13]. In the 2016 family, the companion paper proves positivity of the first matrix of a coefficient pencil for sizes $N \leq 8$ , with real, interlacing roots below $F (p)$ for $p > 1$ . The shifted minors and coefficientwise positivity are supported by the stated finite polynomial computations. The coefficient-moment section gives the definitions and the precise ranges. Positivity for every $N$ and convergence of the largest root remain open. The general moment criteria of Wang and Zhu [15] and Sokal and Walrad [17], Berg’s factorial-power example [16], and the quadrature construction of Golub and Welsch [18] provide the relevant comparisons. Using a recurrence from another family would require a proof that it holds for these coefficients. The root-of-unity method of Krattenthaler, Rochev, Väänänen and Zudilin [14], for example, concerns a separate construction.

Congruences and divided forms

For $Q \in Z [X]$ of degree at most $W$ , the homogenised value $H_{W} (Q) = 2^{W} Q (3 / 2)$ is integral. The companion treats its congruences at powers of $2$ and $3$ , including integer rescaling and endpoint congruences and signed sums, with the rank-one case. Smith normal form counts the residue image [6]. To obtain a small nonzero remainder after division, the count must also distinguish nearby real values from repeated ones.

Let $M$ integral rows $(A_{j}, B_{j})$ have remainders $e_{j} = A_{j} F (3 / 2) - B_{j}$ , and let $D$ be a positive integer. Suppose the subset sums $\sum_{j \in I} (A_{j}, B_{j})$ take $Q$ values in $(Z / D Z)^{2}$ , with at most $k$ subsets giving any fixed real remainder within a residue class. Put $T = \sum_{j} | e_{j} |$ . The least and greatest subset remainders are

\sum_{e_{j} < 0} e_{j} and \sum_{e_{j} > 0} e_{j};

their difference is $T$ . For a positive integer $n$ , divide this whole range into half-open intervals of width $D / n$ , starting at its least value. At most $⌊ n T / D ⌋ + 1$ intervals suffice, including a last interval when the greatest value is an endpoint. If every residue–interval pair contained only one remainder value, it would contain at most $k$ subsets. Hence

\begin{matrix} (16) & 2^{M} > Q k (⌊ \frac{n T}{D} ⌋ + 1) \end{matrix}

forces two different remainders in one residue–interval pair. Subtracting the subset sums and dividing by $D$ gives an integral linear form of nonzero absolute value less than $1 / n$ : the residue condition gives integrality, the interval gives smallness, and the multiplicity bound prevents a zero difference. The quantitative lemma and its proof state the finite-set result independently of Lambert series. Distinct input rows do not rule out equal subset remainders; $k$ bounds repetitions within a residue class. No family of bounds on $Q, T, k$ satisfying (16) for arbitrarily large $n$ at $3 / 2$ is supplied here.

The elementary integer-base method provides a different comparison. Vandehey [10] and Duverney and Tachiya [11] use integer-base expansions. At a noninteger rational base the cleared-tail recurrence has a growing denominator contribution. The clearing and tail-recurrence calculations in the companion paper include the coordinatewise obstruction at $3 / 2$ . They make no assertion about other approximation families. At the function level, Bell and Smertnig [9] exclude Mahler equations for the divisor generating series, while leaving the arithmetic of a single rational argument undecided.

A construction problem.

The unrestricted request for primitive polynomial rows and a signed combination with divided remainder strictly between $0$ and $1 / n$ is equivalent to irrationality of the target; the companion’s section on divided linear forms states and proves the equivalence. A more specific problem must prescribe a family: for the 2004 forms, the question is whether a parameter direction improves $θ^{*}$ . The all-base degree restriction bounds the degree cutoff by $1 / 2$ . It assumes one polynomial family, with common leading constants in its degree, coefficient-height and remainder estimates at every fixed real base $x > 1$ ; errors may depend on $x$ . A cutoff above $1 / 2$ would make remainder decay outpace coefficient growth at a sufficiently large fixed integer base. Consecutive integer row determinants would then tend to zero and eventually vanish, forcing the rational row ratios to stabilise. The remainders would be either zero or bounded away from zero, contradicting nonzero decay. Estimates only at $3 / 2$ do not permit that choice of base.

Under the same hypotheses, the nondecay corollary excludes decay of the denominator-cleared, undivided forms whenever $b < a < b^{2}$ , including $3 / 2$ , without assuming a quadratic degree limit. If that limit exists, their absolute values tend to infinity throughout this strict region. Further division by base-dependent content and families outside these hypotheses remain unclassified, as does equality in the cutoff of Section 4.

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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