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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Exact degrees

The final summand of $A_{n}$ has strictly larger degree than all the others. We use it to determine the leading coefficient as well as the degree. After cancellation, this will give the least common power of the base denominator that clears the pair.

Let $d_{n, k} = \deg (c_{n, k} (X) X^{a_{0} k})$ . The Gaussian factors are monic of degrees $(a_{1} - 1) (k - a_{1})$ and $(β_{n} - k - 1) (k - a_{2})$ . Adding these to the monomial exponent gives

d_{n, k} = e_{n, k} + a_{0} k + (a_{1} - 1) (k - a_{1}) + (β_{n} - k - 1) (k - a_{2}) = \frac{- k^{2} + 80 k n + 3 k - 340 n^{2} - 26 n}{2} .

Taking the difference at consecutive indices, we obtain

d_{n, k + 1} - d_{n, k} = 40 n + 1 - k > 0 (a_{2} \leq k \leq β_{n} - 2),

the inequality because $k \leq β_{n} - 2 = 27 n$ and $27 n < 40 n + 1$ . The summand of top degree is therefore the one at $k = β_{n} - 1$ alone, no cancellation occurs there, and

\begin{matrix} (8) & \begin{matrix} K_{n} := \deg A_{n} = d_{n, β_{n} - 1} = \frac{1091 n^{2} + 81 n + 2}{2}, \\ W_{n} := \deg U_{n} = K_{n} - M_{n} + \sum_{l \leq N} φ (l) - \sum_{2 \leq l \leq N} ν_{n, l} φ (l), \end{matrix} \end{matrix}

the second equality because $\deg D_{N} = \sum_{l \leq N} φ (l)$ and $\deg Ω_{n} = \sum_{2 \leq l \leq N} ν_{n, l} φ (l)$ , and because $U_{n}$ is a polynomial by (7), so subtracting $M_{n}$ from the degree of $D_{N} A_{n} / Ω_{n}$ is legitimate. In particular $U_{n} \neq 0$ . There is also no integer leading-coefficient factor to cancel: the top summand has leading coefficient $(- 1)^{a_{1} + a_{2} + β_{n}} = (- 1)^{n}$ , and both Gaussian factors and $D_{N} / Ω_{n}$ are monic. Thus the leading coefficient of $U_{n}$ is $(- 1)^{n}$ .

The net decrease in degree is

K_{n} - W_{n} = M_{n} - \deg (D_{N} / Ω_{n}) .

Dividing by $X^{M_{n}}$ lowers the degree by $M_{n}$ , but multiplying by the remaining cyclotomic factors increases it by $\deg (D_{N} / Ω_{n})$ . This is why the remainder estimate below involves $K_{n} - W_{n}$ , rather than $M_{n}$ alone.

For a reduced fraction $a / b$ , the unit leading coefficient also gives the exact denominator of $U_{n} (a / b)$ , not just an upper bound. For every prime $p ∣ b$ ,

b^{W_{n}} U_{n} (a / b) \equiv (- 1)^{n} a^{W_{n}} ≢ 0 (\mod p) .

No prime dividing $b$ cancels from the cleared numerator, so the reduced denominator is exactly $b^{W_{n}}$ . In particular every common integer divisor of the cleared row is coprime to $b$ . This excludes cancelling a factor of $b$ from this row, not cancellation at other primes or a saving from a different pair.

The formal arithmetic checks cover the value of $2 M_{n}$ from the source’s formula (16), the degree difference $d_{n, k + 1} - d_{n, k} = 40 n + 1 - k$ , its positivity for $a_{2} \leq k \leq β_{n} - 2$ , and the top degree $2 K_{n} = 1091 n^{2} + 81 n + 2$ . These statements check the polynomial degree calculation, not the analytic estimates used later.

For the second coefficient, fix $n$ and let $x \to \infty$ . For $x \geq 2$ , every finite product in (2) lies between $\prod_{j \geq 1} (1 - 2^{- j}) > 0$ and $1$ , independently of $t$ . Also $\sum_{t \geq 0} x^{- a_{0} t} \leq 2$ . These bounds give $H_{n} (x) = O (1)$ for the whole sum, not just each summand. Also $F (x) = x^{- 1} + O (x^{- 2})$ , because $τ (1) = 1$ and $\sum_{m \geq 2} m x^{- m} = O (x^{- 2})$ . The monic normalising factor at infinity gives $Λ_{n} (x) = O (x^{W_{n} - K_{n}})$ . Since $K_{n} \geq 2$ and $U_{n}$ has leading coefficient $(- 1)^{n}$ ,

V_{n} (x) = U_{n} (x) F (x) - Λ_{n} (x) = (- 1)^{n} x^{W_{n} - 1} + O (x^{W_{n} - 2}) .

The degree formula gives $W_{n} \geq K_{n} - M_{n} = (559 n^{2} + 13 n) / 2 > 0$ . Polynomial integrality was established earlier, so the displayed asymptotic proves the exact statement

\begin{matrix} (9) & \deg V_{n} = W_{n} - 1, lc V_{n} = (- 1)^{n} . \end{matrix}

In particular $V_{n}$ is not the zero polynomial. Applying the same prime-by-prime reduction at its own degree shows that $V_{n} (a / b)$ has reduced denominator exactly $b^{W_{n} - 1}$ . Thus $b^{W_{n}}$ is the least common clearing denominator, although the second coordinate alone needs one fewer power of $b$ . In the common normalisation,

gcd (b^{W_{n}} V_{n} (a / b), b^{W_{n}}) = b .

Indeed $b^{W_{n}} V_{n} (a / b)$ is $b$ times an integer coprime to $b$ . These facts concern the fixed source pair, not all approximants. The degree calculation fixes $n$ and varies $x$ ; the decay calculation fixes $x$ and varies $n$ .

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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