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 rational bracket around $θ^{*}$

Rational bounds for the trigamma differences locate $θ^{*}$ without a numerical approximation to the full sums. Their first terms already place $31 / 4$ inside the region and $3 / 2$ outside it. A sharper enclosure then gives the irrationality-exponent bound $301$ .

For the lower bound, keep only the $k = 0$ term of each of the thirteen differences $ψ_{1} (u_{i}) - ψ_{1} (v_{i})$ . All later terms are positive, so

J \geq \sum_{i = 1}^{13} (\frac{1}{u_{i}^{2}} - \frac{1}{v_{i}^{2}}) = \frac{2015640690251}{25971865920} > \frac{776}{10} .

Since $π > 157 / 50$ and $225 - J < 225$ ,

C_{0} > 266 - \frac{3 (225 - 776 / 10)}{(157 / 50)^{2}} = \frac{5451134}{24649} > \frac{88371}{400} = \frac{81}{200} C_{1},

so $81 / 200 < θ^{*}$ . For the upper bound the thirteen half-open intervals are disjoint and ordered and $ψ_{1}$ is positive and decreasing, so the sum telescopes below its first term:

J < ψ_{1} (1 / 14) = 196 + \sum_{k \geq 1} \frac{1}{(k + 1 / 14)^{2}} < 196 + \frac{π^{2}}{6} < 198 < 225,

whence $C_{0} < 266$ and $θ^{*} < 532 / 1091 < 1 / 2$ . So

\frac{81}{200} < θ^{*} < \frac{1}{2} .

Lean checks the definitions of $C_{0}$ , $C_{1}$ and $θ^{*}$ together with the lower bound on $J$ , rational lower bound on $C_{0}$ , and the two comparisons $81 / 200 < θ^{*}$ and $θ^{*} < 1 / 2$ . It also checks that $31 / 4$ belongs to the region, that every positive integral power does too, and that $3 / 2$ is excluded. These comparisons use the defined constant; the analytic estimates belong to the proof of Theorem 2.2, which Lean also checks. Lean also proves $0.40568302138406054100 \leq θ^{*} \leq 0.40568302138406054104$ and enclosures of $J$ and $C_{0}$ (evidence record, Theorem 2.2). The sharper interval in the preceding proof comes from the stated exact-arithmetic certificate with $256$ summands per trigamma difference. It does not rely on the historical thousand-term decimal evaluation reported in Section 2.7.

The shorter interval above suffices for the bound $301$ but does not certify all digits printed in the cutoff. Lean’s enclosure of $θ^{*}$ implies the following bounds, and a separate integer-arithmetic calculation reproduces them independently of Lean:

\begin{aligned} 0.4056830213840605403 & < θ^{*} < 0.4056830213840605417, \\ 2.4649786835749750334 & < μ < 2.4649786835749750415 . \end{aligned}

Here is the complete error bound used in that calculation. For each of the thirteen intervals, put $f_{u, v} (k) = (k + u)^{- 2} - (k + v)^{- 2}$ . Round each of the first $M = 65536$ positive rational summands down to a multiple of $10^{- 50}$ . Their true sum is between this rounded sum and that sum plus $M 10^{- 50}$ . Since $f_{u, v}$ is positive and decreasing, the remaining tail lies between its integral from $M$ to infinity and that integral plus $f_{u, v} (M)$ . The integral is $(M + u)^{- 1} - (M + v)^{- 1}$ . Summing these rational intervals and using Machin’s alternating-series bounds for $π$ gives the displayed bounds on $θ^{*}$ and its reciprocal. The coefficient $225 - J$ is positive, so the interval endpoints for $C_{0}$ have the claimed order. This verifies the displayed decimal truncations; the program and its recorded result are public. The proofs use the exact definitions and never these decimal values.

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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The rational bracket around 𝜃∗θ∗

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The rational bracket around $θ^{*}$