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

The polynomial forms and their common factors

We construct $U_{n}, V_{n}$ as integral polynomials before evaluating them. Cancelling their common factors at this stage lowers the degree and hence the power of $b$ required at a rational base $a / b$ .

For each $n \geq 1$ , the coefficient formulas below are rational functions of an indeterminate $X$ . We evaluate them at real $x > 1$ and put $q = x^{- 1}$ when using the convergent series. Set

\begin{matrix} a_{0} = 14 n + 1, a_{1} = 12 n + 1, a_{2} = 14 n + 1, \\ β_{n} = 27 n + 2, N = 15 n, M_{n} = 266 n^{2} + 34 n + 1 . \end{matrix}

The symbol $β_{n}$ is the parameter $b$ of [8], the exponent in the lower parameter $q^{b}$ of the Heine series (13) on p. 157; the letter $b$ continues to denote the denominator of the base. In this construction alone, $N = 15 n$ is the cyclotomic cutoff. The exponent $M_{n}$ is the integer of that paper’s (16) on p. 157. Write $(z; q)_{m} = \prod_{j = 0}^{m - 1} (1 - z q^{j})$ for the $q$ -Pochhammer symbol (the shifted $q$ -factorial). Zudilin’s positive remainder is

\begin{matrix} (2) & H_{n} (x) = \sum_{t \geq 0} \frac{(q^{t + 1}; q)_{a_{1} - 1}}{(q; q)_{a_{1} - 1}} \frac{(q; q)_{β_{n} - a_{2} - 1}}{(q^{a_{2} + t}; q)_{β_{n} - a_{2}}} q^{a_{0} t} . \end{matrix}

The denominator length $β_{n} - a_{2}$ agrees with the gamma expression (7) and residues (8) of [8]. The unnumbered display of $R (T)$ on that page instead has length $β_{n} - a_{2} - 1$ ; we use the normalisation fixed by the numbered identities. Then [8] gives $H_{n} = A_{n} F - B_{n}$ with $A_{n}, B_{n} \in Q (X)$ . We need their explicit formulas to compute degrees and bound coefficients. Let ${[\binom{m}{r}]}_{X} = \prod_{j = 1}^{r} (1 - X^{m - r + j}) / (1 - X^{j})$ for $0 \leq r \leq m$ . This Gaussian binomial, or $q$ -binomial, polynomial has degree $r (m - r)$ and nonnegative coefficients summing to $(\binom{m}{r})$ . For $a_{2} \leq k \leq β_{n} - 1$ put

\begin{matrix} (3) & c_{n, k} (X) = (- 1)^{a_{1} + a_{2} + k + 1} X^{e_{n, k}} {[\binom{k - 1}{a_{1} - 1}]}_{X} {[\binom{β_{n} - a_{2} - 1}{β_{n} - k - 1}]}_{X}, \end{matrix}

e_{n, k} = \frac{a_{1} (a_{1} - 1)}{2} - \frac{(β_{n} - a_{2}) (β_{n} - a_{2} - 1)}{2} + \frac{(β_{n} - k) (β_{n} - k - 1)}{2} .

Then

\begin{aligned} (4) & A_{n} (X) & = \sum_{k = a_{2}}^{β_{n} - 1} c_{n, k} (X) X^{a_{0} k}, \\ (5) & B_{n} (X) & = \sum_{k = a_{2}}^{β_{n} - 1} c_{n, k} (X) X^{a_{0} k} (\sum_{l = 1}^{k - a_{1}} \frac{1}{X^{l} - 1} + \sum_{j = 1}^{a_{0} - 1} \frac{X^{- j (k - a_{1})}}{X^{j} - 1}) . \end{aligned}

Put

\begin{matrix} (6) & D_{N} (X) = \prod_{l = 1}^{N} Φ_{l} (X), ν_{n, l} = ω (n / l), Ω_{n} (X) = \prod_{l = 2}^{N} Φ_{l} (X)^{ν_{n, l}}, \end{matrix}

with $Φ_{l}$ the $l$ th cyclotomic polynomial; by Lemma 2.1 every $ν_{n, l}$ is $0$ or $1$ , and $ν_{n, l} = 0$ for $l > 15 n$ because the intervals all begin at or above $1 / 14$ .

Zudilin’s Lemma 7 [8] applies at the parameters above: the tuple is $c = n \cdot (13, 14, 12, 14, 15, 13)$ with maximum $m (c) = 15 n$ and difference $s (c) = n > 0$ , and the source’s (14) on p. 157 holds because $12 n + 1 \leq 14 n + 1$ and $26 n + 2 \leq 27 n + 2 \leq 28 n + 2$ . Its conclusion is the inclusion

\begin{matrix} (7) & U_{n} := X^{- M_{n}} \frac{D_{N}}{Ω_{n}} A_{n} \in Z [X], V_{n} := X^{- M_{n}} \frac{D_{N}}{Ω_{n}} B_{n} \in Z [X] . \end{matrix}

Two points about that citation matter. First, the conclusion used is the one in $Z [X]$ , not integrality at integer arguments: an integer-valued polynomial such as $X (X - 1) / 2$ shows that the two statements differ. The source’s proof supplies the polynomial form. Its Lemma 4 (pp. 157–159) clears the two rational coefficients separately by Gaussian-binomial identities in $Z [X]$ . Lemma 5 (p. 160) makes the normalised series invariant under the order-twelve group generated by Heine’s parameter transformation and the exchange of $a_{1}$ and $a_{2}$ . For the divisibility argument, however, only its order-six subgroup preserving $s > 0$ is used: one application of Heine’s transformation reverses the sign of $s$ . Under that subgroup the normalising factorial product takes the three forms displayed in the source. Comparing their cyclotomic valuations gives the maximum of three terms in $ω$ . Lemma 7 (p. 161) removes the resulting common cyclotomic factors by polynomial divisibility; the divisor $Ω_{n}$ is monic. These steps take place before the integer specialisation in (24) on p. 162.

Second, the module inclusion gives a polynomial coefficient pair, and uniqueness identifies it with the displayed cancelled source pair. If two pairs differed, subtracting their identities would express $F$ as a rational function of $X$ . This is impossible: as $h ↓ 0$ , splitting the original Lambert sum $F (e^{h}) = \sum_{m \geq 1} (e^{h m} - 1)^{- 1}$ at $L = ⌊ 1 / h ⌋$ gives an initial sum $h^{- 1} \sum_{m \leq L} m^{- 1} + O (L)$ , since $y^{- 1} - 1 \leq (e^{y} - 1)^{- 1} \leq y^{- 1}$ for $0 < y \leq 1$ . The remaining sum is at most $e^{- h (L + 1)} / ((1 - e^{- 1}) (1 - e^{- h})) = O (h^{- 1})$ . Hence $F (e^{h}) = h^{- 1} \log (1 / h) + O (h^{- 1})$ , so $(X - 1) F (X) \to \infty$ and $(X - 1)^{2} F (X) \to 0$ as $X ↓ 1$ , which no rational function does. Thus the source’s module inclusion yields the specific pair in (7), not merely some unspecified polynomial representation of the same function. This functional nonrationality says nothing about an individual value. For $x > 1$ , write the cancelled remainder as

Λ_{n} (x) = U_{n} (x) F (x) - V_{n} (x) = x^{- M_{n}} \frac{D_{N} (x)}{Ω_{n} (x)} H_{n} (x) .

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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