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 region $b^{μ} < a$ and the base $31 / 4$

The rational-base calculation compares a gain in the remainder with the cost of making its coefficients integral. If the original coefficient has degree $K_{n}$ and the cancelled coefficient has degree $W_{n}$ , the leading balance is

\begin{matrix} (1) & b^{W_{n}} (a / b)^{- (K_{n} - W_{n})} = b^{K_{n}} a^{- (K_{n} - W_{n})} . \end{matrix}

The polynomial identities and remainder estimates below justify this balance, including its lower-order factors. With $K_{n} / n^{2} \to C_{1}$ and $(K_{n} - W_{n}) / n^{2} \to C_{0}$ , the cleared remainder tends to zero when $C_{1} \log b < C_{0} \log a$ . The degree drop has two effects: it improves the real remainder and changes the denominator needed to make its coefficients integral. Both effects concern the same cancelled form. This is why one must calculate the degree before rational evaluation, rather than borrow a denominator bound from the uncancelled family.

We evaluate Zudilin’s 2004 forms at $a / b$ after cancelling their common polynomial factors. His direction $(14, 12, 14; 27)$ [8] gives the constants below; the comparison between decay and the remaining degree determines the irrationality region:

C_{1} = (14 + 12 + 14) 27 - \frac{1}{2} (12^{2} + 14^{2} + 27^{2}) = \frac{1091}{2}, C_{0} = 266 - \frac{3}{π^{2}} (225 - J),

J = \sum_{i = 1}^{13} (ψ_{1} (u_{i}) - ψ_{1} (v_{i})), ψ_{1} (x) = \sum_{k \geq 0} \frac{1}{(k + x)^{2}},

where $ψ_{1}$ is the trigamma function, used only for positive arguments, and the thirteen half-open intervals $[u_{i}, v_{i})$ are $[1 / 14, 1 / 12)$ , $[1 / 7, 1 / 6)$ , $[3 / 14, 1 / 4)$ , $[2 / 7, 1 / 3)$ , $[5 / 14, 2 / 5)$ , $[3 / 7, 7 / 15)$ , $[1 / 2, 8 / 15)$ , $[4 / 7, 3 / 5)$ , $[9 / 14, 2 / 3)$ , $[5 / 7, 11 / 15)$ , $[11 / 14, 4 / 5)$ , $[6 / 7, 13 / 15)$ , $[13 / 14, 14 / 15)$ . The intervals are disjoint and lie in $[1 / 14, 1)$ , so $0 \leq J \leq ψ_{1} (1 / 14) - ψ_{1} (1) < 196$ : in the defining series, $ψ_{1} (1 / 14) = 196 + \sum_{k \geq 1} (k + 1 / 14)^{- 2} < 196 + ψ_{1} (1)$ . Since $π > 3$ , it follows that $191 < C_{0} < 266 < C_{1} / 2$ . In particular, the reciprocal constants below are positive without using decimal approximations. The accompanying rational certificate gives

\begin{aligned} 77.94318445520543 & < J < 77.94318449473569, \\ 221.30008815898543 & < C_{0} < 221.30008817100119, \\ 0.40568302137302 & < θ^{*} := C_{0} / C_{1} < 0.40568302139506 . \end{aligned}

Put $μ = 1 / θ^{*}$ . These parameter ratios, intervals and constants, including $C_{1} / C_{0} = 2.46497868 \dots$ , occur in [8]; that ratio is the integer-base exponent bound in its Theorem 1, not a new optimised constant. The displayed interval endpoints are rounded outwards from rational bounds; no digits beyond their certified accuracy are required.

The proof below needs $J$ in its combinatorial form as well. Put

ω (ξ) = max {0, ⌊ 14 ξ ⌋ + ⌊ 13 ξ ⌋ - ⌊ 12 ξ ⌋ - ⌊ 15 ξ ⌋, 2 ⌊ 14 ξ ⌋ - ⌊ 13 ξ ⌋ - ⌊ 15 ξ ⌋},

the weight that [8] attaches to the six-tuple $c = (13, 14, 12, 14, 15, 13)$ for the chosen parameter ratios; it gives the exponents $ν_{l} = ω (n / l)$ of the source’s (22) and enters its limit (26). Zudilin lists the support of $ω$ in the same place; the next lemma verifies that list by exact evaluation. Since $14 + 13 = 12 + 15$ and $2 \cdot 14 = 13 + 15$ , each floor difference is unchanged by $ξ \mapsto ξ + 1$ . Thus $ω$ has period $1$ , and the lemma also determines every value $ω (n / l)$ needed for the cyclotomic exponents.

Lemma 2.1 (the weight is an indicator). On $[0, 1)$ the function $ω$ takes only the values $0$ and $1$ , and $ω = 1$ exactly on the union of the thirteen half-open intervals listed above.

Proof. Each of the three expressions inside the maximum is constant on every interval between consecutive elements of ${k / c : c \in {12, 13, 14, 15}, 0 \leq k \leq c}$ , since those are the only points at which one of the four floor functions changes. There are $48$ such intervals in $[0, 1)$ ; evaluating $ω$ at the midpoint of each gives the value $1$ on the thirteen listed half-open intervals and $0$ elsewhere, and the thirteen listed intervals are unions of consecutive cells. Each floor function is right-continuous, so a cell has the same value at its left endpoint as at its midpoint. This also verifies the half-open endpoint convention. All evaluations use exact rational arithmetic. ◻

Lemma 2.1 and termwise integration of the positive series $- ψ_{1}^{'} (ξ) = \sum_{k \geq 0} 2 (k + ξ)^{- 3}$ give the two forms of $J$ used below,

J = \int_{0}^{1} ω (ξ) \sum_{k \geq 0} \frac{2}{(k + ξ)^{3}} d ξ = \sum_{i = 1}^{13} (ψ_{1} (u_{i}) - ψ_{1} (v_{i})),

the first being the integral $\int_{0}^{1} ω d (- ψ_{1})$ of [8]. Since $ω$ vanishes near $0$ , both are finite.

For $b = 1$ the logarithmic ratio is zero, so the condition includes every integer base. For $b > 1$ it requires $a > b^{2.46497868 \dots}$ , substantially more than $a > b$ . There are still infinitely many admissible coprime numerators for each fixed denominator. The base $31 / 4$ is included, whereas $3 / 2$ is excluded. Taking positive integral powers does not change the ratio. At equality with the cutoff, the estimates give no conclusion.

Theorem 2.2 (rational-base region). Let $a > b \geq 1$ be coprime integers with

b^{μ} < a, equivalently \frac{\log b}{\log a} < θ^{*} = 0.40568302138406054 \dots

Then $F (a / b) = \sum_{m \geq 1} ((a / b)^{m} - 1)^{- 1}$ is irrational.

The proof has three steps. We cancel the common polynomial factors using Zudilin’s integrality argument [8]. We then compute the degree of the remaining polynomials, using his cyclotomic limits and the reciprocal-interval method of [7]. Finally, we evaluate at $a / b$ and show that the positive remainder still decays after multiplication by the required power of $b$ . These steps occupy Sections 2.1–2.4. The construction, direction and constants are Zudilin’s; the rational specialisation and its degree and remainder estimates are proved below.

The polynomial forms and their common factors

Exact degrees

The cyclotomic limit

Positive linear forms with integer coefficients

The rational bracket around 𝜃∗θ∗

Earlier criteria and changes of direction

Finite calculations

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 region 𝑏𝜇 <𝑎bμ<a and the base 31/431/4

About this paper

The region $b^{μ} < a$ and the base $31 / 4$