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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A congruence relation with nonzero remainder

Fixing the approximation family is not enough: the residue map must also use a fixed normalisation. Polynomial coefficient content, content of an evaluated row, and content of a combined row are three different quantities. We keep them separate as follows. For each $n$ , let $W_{n}$ be a common degree bound and let

(U_{n, j}, V_{n, j}, R_{n, j})_{j < M_{n}}

be the chosen forms, with $U_{n, j}, V_{n, j} \in Z [X]$ of degree at most $W_{n}$ and

R_{n, j} (t) = U_{n, j} (t) F (t) - V_{n, j} (t) (t > 1) .

This identity uses the same normalisation as the evaluated rows and all subsequent residue equations. Fix all divisions before forming the residue map. Require each polynomial pair to have coefficient gcd $1$ , by dividing its common integer coefficient factor if necessary. This is different from making the two evaluated integers coprime, as required in Problem 10.1. The two operations need not preserve the same polynomial family. Coefficient content $1$ alone gives no coprimality with $6$ after evaluation: at $W = 1$ , the pairs $(X, 3)$ and $(2 X - 1, 2)$ have coefficient content $1$ but give the integer rows $(3, 6)$ and $(4, 4)$ , respectively.

The gcd

gcd (H_{W_{n}} (U_{n, j}), H_{W_{n}} (V_{n, j}))

of one evaluated row can differ from the gcd of the final sum. Neither is divided out without also dividing the corresponding remainder and measuring the height after that division. If, in addition, the coefficient of $X^{W_{n}}$ in $U_{n, j}$ and the constant coefficient of $V_{n, j}$ are both units, Proposition 5.3 makes this row gcd coprime to $6$ . These endpoint assumptions are extra conditions, not consequences of coefficient primitivity. Dividing by such a gcd preserves divisibility by every power of $2$ and $3$ for that individual row. It need not preserve a relation with a fixed coefficient vector when different rows are divided by different contents. For example, at $W = 1$ the pairs $(X - 1, X - 1)$ and $(X + 1, X + 1)$ give rows $(1, 1)$ and $(5, 5)$ . Their sum is zero modulo $6$ , whereas the sum of their primitive normalisations is $(2, 2)$ , not zero modulo $6$ . Both pairs satisfy the two unit conditions just stated. Each division is invertible modulo $6$ , but different rows are multiplied by different units, so their sum need not stay zero. The residue map must therefore be formed again after row-by-row normalisation.

Dividing a specialised row by its content need not preserve the chosen polynomial family, and lifting the divided row back is a constrained problem. For fixed $W$ , the map $P \mapsto H_{W} (P)$ on integer polynomials of degree at most $W$ is surjective onto $Z$ , since $3^{W}$ and $2^{W}$ are coprime, so a lift always exists. The issue is not an arbitrary exponential height bound, as the construction above shows. It is whether the lift belongs to the chosen approximation family and satisfies that family’s remainder identity and quantitative height estimate. With $W = 1$ , for instance, $(X + 1, X + 1)$ specialises to $(5, 5)$ , whose primitive normalisation $(1, 1)$ lifts to $(X - 1, X - 1)$ , but not to an integer scalar multiple of the original pair. Any lift used below is therefore supplied together with its degree bound, its height bound and its exact remainder.

For target depths $R_{n}, S_{n}$ , let $J_{n} (λ)$ be the vector of four residues of the pair $\sum_{j} λ_{j} (U_{n, j}, V_{n, j})$ , and define

C_{n} = {λ \in {- 1, 0, 1}^{M_{n}} \ {0} : J_{n} (λ) = 0},

\begin{aligned} K_{n}^{poly} & = {λ \in {- 1, 0, 1}^{M_{n}} : \sum_{j} λ_{j} U_{n, j} = 0, \sum_{j} λ_{j} V_{n, j} = 0}, \\ K_{n}^{rem} & = {λ \in {- 1, 0, 1}^{M_{n}} : \sum_{j} λ_{j} R_{n, j} (3 / 2) = 0} . \end{aligned}

The second set consists of signed relations whose remainder vanishes at $3 / 2$ . It also contains every relation with an identically zero remainder function. These are sets of restricted coefficient vectors, not assertions that the signed cube is a vector space.

The exact pigeonhole condition is

\begin{matrix} (26) & 2^{M_{n}} > 3^{2 R_{n}} 2^{2 S_{n}}, equivalently M_{n} > 2 R_{n} \log_{2} 3 + 2 S_{n} . \end{matrix}

Thus the least integer number of rows satisfying this counting test is

⌊ 2 R \log_{2} 3 + 2 S ⌋ + 1.

The checked condition $M \geq 4 R + 2 S$ for $R > 0$ is a convenient sufficient corollary, not the exact threshold. Applying Cauchy–Schwarz to the sizes of the residue classes shows that there are at least

\begin{matrix} (27) & \frac{1}{2} (\frac{2^{2 M_{n}}}{3^{2 R_{n}} 2^{2 S_{n}}} - 2^{M_{n}}) \end{matrix}

unordered pairs of distinct subsets with equal residues. It would therefore suffice to prove that fewer than this many pairs give a zero polynomial combination or a zero remainder at $3 / 2$ .

Theorem 5.8 supplies only $C_{n} \neq \emptyset$ ; it does not rule out zero polynomial pairs or zero real remainders. For a useful approximation family, bounding the multiplicities in (27) remains a possible way to obtain nonvanishing. Without restrictions on the family, however, nonvanishing alone is elementary.

Example 10.2 (nonvanishing without decay). For $n \geq 1$ , set $W_{n} = R_{n} = S_{n} = n^{2}$ and $M_{n} = 6 n^{2}$ . Take

\begin{aligned} (U_{n, 0}, V_{n, 0}) & = ((1 + 2^{W_{n}}) X^{W_{n}}, 1), \\ (U_{n, j}, V_{n, j}) & = (X^{W_{n}}, 1) (1 \leq j < M_{n}), \end{aligned}

and define $R_{n, j} (t) = U_{n, j} (t) F (t) - V_{n, j} (t)$ for $t > 1$ . All pairs have coefficient gcd $1$ and primitive evaluated rows. The coefficient vector $(1, - 1, 0, \dots, 0)$ belongs to $C_{n} \ (K_{n}^{poly} \cup K_{n}^{rem})$ , but the divided integer form equals $F (3 / 2)$ for every $n$ .

Indeed, the evaluated rows are $(3^{W_{n}} + 6^{W_{n}}, 2^{W_{n}})$ and $(3^{W_{n}}, 2^{W_{n}})$ ; their first coordinates are odd, so both rows are primitive. Their difference is $(6^{W_{n}}, 0)$ , while the polynomial difference is $(2^{W_{n}} X^{W_{n}}, 0)$ . Thus all four residues vanish. The unscaled remainder difference is $3^{W_{n}} F (3 / 2) > 0$ . Clearing the evaluation denominators multiplies it by $2^{W_{n}}$ , giving $6^{W_{n}} F (3 / 2)$ . Dividing this cleared integer form by $3^{R_{n}} 2^{S_{n}} = 6^{W_{n}}$ leaves the same positive constant $F (3 / 2)$ . The counting condition (26) holds because $\log_{2} 3 < 2$ .

Example 10.2 meets the two nonvanishing requirements in Problem 10.1 without any irrationality assumption. What it fails is the smallness condition. The useful question is therefore to obtain nonvanishing and decay in the same prescribed approximation family, not merely to exhibit some family with a nonzero congruence relation.

The range of subset sums.

The lattice after evaluation.

The lattice spanned by the actual tails.

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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