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

Power comparisons and Hankel determinants

Three integer comparisons will be used to estimate residue counts and scalar exponents. The first gives $\log 3 / \log 2 < 65 / 41$ and shows that $65$ is the least integer $q$ with $3^{41} < 2^{q}$ ; the inequality $2^{129} < 3^{82}$ checks the failure of the subset count at depth $41$ one row earlier. They determine numerical thresholds, leaving the coefficient and remainder hypotheses to be checked for any family of approximants. Readers interested first in the determinant asymptotic may begin at Section 3.1.

Theorem 3.1 (sharp power bracket). One has

2^{64} < 3^{41} < 2^{65} .

Consequently

\frac{41}{65} < \frac{\log 2}{\log 3}, \frac{\log 3}{\log 2} < \frac{65}{41} .

Proof. Direct evaluation gives

\begin{aligned} 2^{64} & = 18446744073709551616, \\ 3^{41} & = 36472996377170786403, \\ 2^{65} & = 36893488147419103232. \end{aligned}

Taking logarithms of the upper bound gives $41 \log 3 < 65 \log 2$ . Dividing this inequality by $65 \log 3$ and by $41 \log 2$ , respectively, gives the two stated logarithmic bounds; both divisors are positive. ◻

The upper power bound gives $\log 2 / \log 3 > 41 / 65$ , a sharper lower bound than $81 / 200$ . In particular, $41 / 65 - 2 / 5 = 3 / 13$ . Combining this with $1 / 2 - 1 / π^{2} < 2 / 5$ gives the first gap below; the same logarithmic lower bound is then compared with the rectangular expression.

For $ρ \geq 0$ and $σ \geq 1 + ρ$ , define

Θ_{HP} (ρ, σ) = \frac{(1 + ρ^{2}) / 2 + σ - 3 σ^{2} / π^{2}}{(1 + ρ)^{2} / 2 + σ (1 + ρ) + (1 + ρ^{2}) / 2 + σ} .

The denominator is positive on this domain.

Corollary 3.2 (gaps between the stated logarithmic thresholds). For every $ρ, σ \in R$ with $0 \leq ρ$ and $1 + ρ \leq σ$ ,

\frac{3}{13} < \frac{\log 2}{\log 3} - (\frac{1}{2} - \frac{1}{π^{2}}), \frac{3}{13} < \frac{\log 2}{\log 3} - Θ_{HP} (ρ, σ),

where $Θ_{HP}$ is the rectangular exponent threshold. Moreover

\frac{\log 3 / \log 2 - 1}{3} < \frac{8}{41} .

Proof. The first inequality follows from $41 / 65 - 2 / 5 = 3 / 13$ , Theorem 3.1, and $1 / 2 - 1 / π^{2} < 2 / 5$ . The polynomial calculation in Section 10 proves $Θ_{HP} (ρ, σ) \leq 1 / 2 - 1 / π^{2}$ on the stated domain, which gives the second inequality. The final inequality is a direct rearrangement of $\log 3 / \log 2 < 65 / 41$ . ◻

The next inequalities compare two proposed savings in the denominator exponent with the exponent $4 N^{3} - 3 N^{2}$ before division. Here $E$ denotes the exponent saved. The statement assumes the bounds $E \leq N^{3} - N$ or $E \leq 2 N^{3} - N$ ; it does not derive them for a polynomial family. Under either bound, the saving is less than $39 / 41$ of the original exponent. Thus these bounds alone cannot justify the reduction required by this model.

Theorem 3.3 (bounds for two proposed degree savings). For every integer $N > 0$ ,

41 (N^{3} - N) < 39 (4 N^{3} - 3 N^{2}) .

For every integer $N \geq 2$ ,

41 (2 N^{3} - N) < 39 (4 N^{3} - 3 N^{2}) .

Hence the same strict inequalities hold with the left side replaced by $41 E$ whenever, respectively, $E \leq N^{3} - N$ or $E \leq 2 N^{3} - N$ .

Proof. After subtraction, the first inequality is

N (115 N^{2} - 117 N + 41) > 0,

whose quadratic factor is $115 (N - 1)^{2} + 113 (N - 1) + 39$ , positive for $N \geq 1$ . The second becomes

N (74 N^{2} - 117 N + 41) > 0.

Its quadratic factor is $74 (N - 2)^{2} + 179 (N - 2) + 103$ , positive for $N \geq 2$ . The assertions for $E$ follow by monotonicity. ◻

An application of these degree bounds must separately prove that its proposed divisor satisfies one of the stated bounds. These inequalities do not rule out additional factors caused by cancellation in a determinant, a different choice of integral forms, or another proof of irrationality at $3 / 2$ .

The sharp 𝑞q-order of the normalised Hankel determinant

The size of 𝑉∗𝑁VN∗ at a fixed base

Positive Hankel determinants and a rational special value

Coefficient moments and cyclotomic content

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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