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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The lattice spanned by the actual tails.

The formulas above apply to any rank-two lattice. For the rows that the approximation problem actually supplies, the invariant can be computed outright, and it is the smallest one compatible with the base. Let $a > b \geq 1$ be coprime, write the partial sum $S_{m} (b / a) = \sum_{r \leq m} (b / a)^{r} / (1 - (b / a)^{r})$ in lowest terms as $P_{m} / Q_{m}$ with $Q_{m} > 0$ , and set $P_{0} = 0$ , $Q_{0} = 1$ . The primitive integer row of the $m$ th tail is $(Q_{m}, P_{m})$ , which represents $Q_{m} F (a / b) - P_{m}$ .

Proposition 10.3 (the prefix lattice of the tails). Every $Q_{m}$ is coprime to $a b$ , and $b$ divides every $P_{m}$ . For every prefix containing $m = 0$ and $m = 1$ ,

{span}_{Z} {(Q_{m}, P_{m}) : 0 \leq m < M} = Z \times b Z (M \geq 2),

so the Smith invariants are $1, b$ , the gcd of the $2 \times 2$ minors is $b$ , and the image modulo $D$ has cardinality $D^{2} / gcd (b, D)$ . Multiplying an individual tail by a nonzero rational weight leaves its primitive row unchanged up to sign.

Proof. A common denominator of $S_{m} (b / a)$ is a product of the integers $a^{r} - b^{r}$ for $r \leq m$ , each coprime to $a b$ ; over that denominator every summand has numerator divisible by $b^{r}$ . Reduction to lowest terms removes no prime factor of $b$ , which gives the first two assertions. The rows at $m = 0, 1$ are $(1, 0)$ and $(a - b, b)$ , and subtracting $a - b$ copies of the first from the second gives $(0, b)$ ; every later row already lies in $Z \times b Z$ by the first assertions. The Smith and image statements are the specialisation of the formulas above at $g = b$ . Finally the rational line through a primitive pair meets $Z^{2}$ in exactly the multiples of that pair, so clearing denominators in a nonzero rational multiple and dividing by the coordinate gcd returns the same pair up to sign. ◻

At $3 / 2$ the first three partial sums are $0$ , $2$ and $14 / 5$ , with rows $(1, 0)$ , $(1, 2)$ and $(5, 14)$ . Their span is already $Z \times 2 Z$ . There is no three-adic collapse, and the only local saving is the single factor of two that the image formula displays. For a consecutive window containing $m = s$ and $m = s + 1$ and no earlier row, every pairwise minor is divisible by $b^{s + 1}$ . The gcd has valuation $(s + 1) v_{p} (b)$ at each prime $p ∣ b$ and valuation zero at each prime $p ∣ a$ . The adjacent-row identity

Q_{s} P_{s + 1} - P_{s} Q_{s + 1} = \frac{Q_{s} Q_{s + 1} b^{s + 1}}{a^{s + 1} - b^{s + 1}}

attains both, since its denominator is a unit at every prime dividing $a b$ .

Rescaling an individual tail and then making its row primitive returns the same row up to sign. Thus it preserves the lattice spanned by the full family. Integer combinations remain in that lattice, but a selected family of combinations need not span all of it and can have a larger minor gcd. At $3 / 2$ , write $u_{0} = (1, 0)$ and $u_{1} = (1, 2)$ . The primitive rows $u_{0}$ and $3 u_{1} - 2 u_{0} = (1, 6)$ span $Z \times 6 Z$ , with minor gcd $6$ , although $u_{0}, u_{1}$ span $Z \times 2 Z$ . The selection replaces the original lattice by an index-three sublattice, accounting for the change from $2$ to $6$ . Containment therefore preserves divisibility by the original gcd, not equality of gcds or endpoint valuations. The prefix and window equalities above concern the full indicated families.

Rational combinations can also leave the original lattice: the half-sum of $u_{0}, u_{1}$ is the primitive row $(1, 1)$ , whose minor against $u_{0}$ is $1$ . Coefficient denominators must then be accounted for; likewise, primitive normalisation after taking combinations requires a new lattice calculation. These operations may change the minor gcd in either direction. None by itself controls the real error: a useful selected family must still have nonzero remainders small enough after the proposed division.

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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