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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In this paper

A restriction on Mahler functional equations

For the divisor generating series $L (z) = \sum_{n \geq 1} τ (n) z^{n}$ , the classification of Bell and Smertnig excludes a $k$ -Mahler equation for every $k \geq 2$ [27]. We first explain its application to $τ$ . We then prove the weaker simultaneous $2$ / $3$ statement by combining the Adamczewski–Bell theorem with the functional nonrationality from Section 2.1. The cited 2026 works have the status of the identified preprints. These functional statements leave the arithmetic nature of an individual rational-base value undecided.

To apply the classification, note that $τ$ is multiplicative: for coprime integers $m, n$ , each divisor of $m n$ has a unique factorisation into a divisor of $m$ and a divisor of $n$ . Suppose that $L$ were $k$ -Mahler. The classification would give a prime $p$ , an integer $r \geq 0$ and an eventually periodic function $χ$ with $τ (m) = m^{r} χ (m)$ whenever $p ∤ m$ . For a prime $ℓ \neq p$ , this forces

χ (ℓ^{j}) = \frac{j + 1}{ℓ^{j r}} (j \geq 0) .

If $r = 0$ , these values are unbounded. If $r > 0$ , they are nonzero and tend to zero. Both alternatives contradict the finite range of an eventually periodic function.

Here the ambient space is $Q ((z))$ , the field of formal Laurent series, viewed as a vector space over $Q (z)$ . Stability means that substituting $z^{k}$ for $z$ sends each member of the subspace back into that subspace; this substitution is not a $Q (z)$ -linear map.

Proposition 10.5 (no finite simultaneous $2 / 3$ -system). Let

L (z) = \sum_{n \geq 1} \frac{z^{n}}{1 - z^{n}} .

There is no finite-dimensional $Q (z)$ -vector space that contains $L$ and is stable under both $z \mapsto z^{2}$ and $z \mapsto z^{3}$ .

Proof. Suppose $V$ were such a space, of dimension $d$ . Stability under $z \mapsto z^{2}$ places the $d + 1$ elements $L (z), L (z^{2}), \dots, L (z^{2^{d}})$ in $V$ , so they are linearly dependent over $Q (z)$ ; clearing denominators gives polynomials $P_{0}, \dots, P_{d}$ , not all zero, with $\sum_{i = 0}^{d} P_{i} (z) L (z^{2^{i}}) = 0$ .

Linear dependence need not give a nonzero coefficient of the unshifted function, as a Mahler equation requires. One cannot simply replace $z$ by $z^{1 / 2^{j}}$ , since the polynomial coefficients need not have exponents divisible by $2^{j}$ . Instead, separate their exponent classes. Let $j$ be the least index with $P_{j} \neq 0$ . Write each polynomial uniquely as $P_{i} (z) = \sum_{r = 0}^{2^{j} - 1} z^{r} Q_{i, r} (z^{2^{j}})$ . Every series $L (z^{2^{i}})$ with $i \geq j$ has exponents divisible by $2^{j}$ , so the relation splits by exponent residues modulo $2^{j}$ . Choose $r$ with $Q_{j, r} \neq 0$ and put $u = z^{2^{j}}$ . The corresponding relation is

\sum_{i = j}^{d} Q_{i, r} (u) L (u^{2^{i - j}}) = 0,

with a nonzero coefficient of $L (u)$ . Thus $L$ is $2$ -Mahler. The same argument with $3$ in place of $2$ makes it $3$ -Mahler. Since $2$ and $3$ are multiplicatively independent, a theorem of Adamczewski and Bell [16] then forces $L$ to be a rational function. Since $L (z) = F (1 / z)$ for $0 < z < 1$ , this contradicts the nonrationality of $F$ proved in Section 2.1. Rivin’s periodic-coefficient corollary [24] gives the same nonrationality conclusion. ◻

Proposition 10.5 uses no property of the point $2 / 3$ : the obstruction is functional and appears before regularity at a particular point is considered. For this scalar function, Bell and Smertnig’s single-base classification already implies the proposition. The proof above instead derives it from simultaneous closure and elementary functional nonrationality, using the Adamczewski–Bell theorem. A construction using additional functions or functional relations must specify those functions and its closure conditions; the single-base statement is not an obstruction to every approximation method. The classification and the Adamczewski–Bell theorem are cited, not proved here; the needed nonrationality has the elementary proof given earlier.

The Lean proof of Proposition 10.5 (no finite simultaneous $2 / 3$ -system) uses neither cited theorem, and it uses only stability under $z \mapsto z^{2}$ . As in the first step of the proof above, that stability gives polynomials $P_{0}, \dots, P_{d}$ , not all zero, with $\sum_{i = 0}^{d} P_{i} (z) L (z^{2^{i}}) = 0$ , and this relation also holds between the values of these functions on the open unit disc. If $ζ$ is a primitive $n$ th root of unity, then $(1 - r) L (r ζ) / \log (1 / (1 - r)) \to 1 / n$ as $r ↑ 1$ . These limits, taken at roots of unity of suitable orders, force every $P_{i}$ to vanish. The same argument excludes, for every $k \geq 2$ , a finite-dimensional space containing $L$ and stable under $z \mapsto z^{k}$ alone (single-base statement).

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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