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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Positive linear forms with integer coefficients

We now specialise at $x = a / b > 1$ . Put $q = 1 / x$ and $P_{q} = (q; q)_{\infty}$ . Every finite $q$ -Pochhammer product in (2) has factors $1 - q^{j}$ with $j \geq 1$ , so it lies between $P_{q}$ and $1$ . Each of the two ratios therefore lies between $P_{q}$ and $P_{q}^{- 1}$ . Summing the resulting geometric bounds gives

P_{q}^{2} \leq H_{n} (x) \leq \frac{P_{q}^{- 2}}{1 - q^{a_{0}}} \leq \frac{P_{q}^{- 2}}{1 - q},

whence $H_{n} (x) > 0$ and $\log H_{n} (x) = O_{x} (1)$ uniformly in $n$ . The cyclotomic factors are positive on $(1, \infty)$ , so the cancelled remainder $Λ_{n} (x)$ is positive as well, by (7). By (8) and (9) both polynomial degrees are at most $W_{n}$ , so

{\hat{U}}_{n} = b^{W_{n}} U_{n} (a / b), {\hat{V}}_{n} = b^{W_{n}} V_{n} (a / b)

are integers, and ${\hat{Λ}}_{n} := b^{W_{n}} Λ_{n} (a / b) = {\hat{U}}_{n} F (a / b) - {\hat{V}}_{n}$ is positive and lies in $Z F (a / b) + Z$ . This follows by multiplying each polynomial value by $b^{W_{n}}$ . The integer coefficients and the earlier cancellation of a common cyclotomic factor are separate inputs. Combining the size estimates,

\begin{aligned} \log {\hat{Λ}}_{n} & = W_{n} \log b - M_{n} \log x + (\sum_{l \leq N} φ (l) - Σ_{n}) \log x + O_{x} (n) \\ = K_{n} \log b - (K_{n} - W_{n}) \log a + o (n^{2}), \end{aligned}

the second equality by $\log x = \log a - \log b$ and (8). With (10) this proves

\begin{matrix} (12) & lim_{n \to \infty} \frac{\log {\hat{Λ}}_{n}}{n^{2}} = C_{1} \log b - C_{0} \log a . \end{matrix}

Proof of Theorem 2.2. The hypothesis $\log b / \log a < θ^{*} = C_{0} / C_{1}$ makes the right side of (12) negative, so ${\hat{Λ}}_{n} > 0$ and ${\hat{Λ}}_{n} \to 0$ . Suppose $F (a / b) = P / Q$ with integers $P$ and $Q \geq 1$ . Then $Q {\hat{Λ}}_{n} = P {\hat{U}}_{n} - Q {\hat{V}}_{n}$ is a positive integer for every $n$ and tends to $0$ , which is impossible. ◻

Above the threshold (12) says that these same homogenised forms grow like $e^{c n^{2}}$ with $c > 0$ , and at equality the limit decides nothing. Both statements are about the displayed family.

The hypothesis $b \geq 1$ admits $b = 1$ , where the statement reduces to the known integer-base theorem. Coprimality fixes the reduced representation of the base. It is used in the exact-denominator assertion above, but is not needed for the sufficient irrationality implication once the displayed logarithmic inequality holds. Negative bases are excluded, because the proof uses positivity for real bases greater than $1$ .

Theorem 2.3 (the base $31 / 4$ outside the Bundschuh–Väänänen region). $F (31 / 4)$ is irrational, and so is $F ((31 / 4)^{r})$ for every integer $r \geq 1$ . Here

\frac{\log 4}{\log 31} = 0.4036981731641997 \dots < \frac{81}{200} < θ^{*},

while

4^{μ} = 30.483515 \dots < 31 < 4^{μ_{BV}} = 32.369642 \dots

with $μ_{BV} = 2 π^{2} / (π^{2} - 2) = 2.508284761994 \dots$ , so $31 / 4$ lies outside the region $\log b / \log a < 1 / 2 - 1 / π^{2} = 0.3986788163576622 \dots$ of [6] and inside the region of Theorem 2.2.

Proof. The comparison $\log 4 / \log 31 < 81 / 200$ is the integer certificate $4^{200} < 31^{81}$ , which Lean checks, and Lean also checks that it places $31 / 4$ in the region $\log b / \log a < 81 / 200$ . The ratio $\log b / \log a$ is invariant under $(a, b) \mapsto (a^{r}, b^{r})$ , which places every power of $31 / 4$ in the same region, and $(31^{r}, 4^{r})$ are coprime. Lean also checks that $31 / 4$ lies outside the earlier region. That exclusion also has a two-line rational certificate: $31^{2} < 4^{5}$ gives $\log 4 / \log 31 > 2 / 5$ , and $π^{2} < 10$ gives $1 / 2 - 1 / π^{2} < 2 / 5$ , so

\frac{1}{2} - \frac{1}{π^{2}} < \frac{2}{5} < \frac{\log 4}{\log 31} < \frac{81}{200} < θ^{*} .

The remaining comparison $81 / 200 < θ^{*}$ is proved in Section 2.5. Theorem 2.2 then applies. ◻

Corollary 2.4 (an irrationality measure uniform over powers). For coprime $a > b \geq 1$ with $θ = \log b / \log a < θ^{*}$ and every integer $r \geq 1$ ,

μ_{i r r} (F ((a / b)^{r})) \leq \frac{1 - θ}{θ^{*} - θ} .

Here $μ_{i r r} (ξ)$ is the supremum of the exponents $ν$ for which $| ξ - p / q | < q^{- ν}$ has infinitely many reduced rational solutions. In particular, $μ_{i r r} (F ((31 / 4)^{r})) < 301$ for every $r \geq 1$ .

Proof. For the irrationality exponent, bound $A_{n}$ in (4): its $O (n)$ Laurent monomials multiply two Gaussian binomial polynomials, each with nonnegative coefficients summing to at most $2^{27 n + 2}$ . Thus its absolute coefficient sum is $\exp (O (n))$ , and its largest exponent $K_{n}$ gives $| A_{n} (x) | \leq x^{K_{n}} \exp (O (n))$ for fixed $x > 1$ . The cyclotomic estimate in Section 2.4 bounds the multiplier $x^{- M_{n}} D_{N} (x) / Ω_{n} (x)$ by $x^{W_{n} - K_{n}} \exp (O_{x} (n))$ . Therefore

| U_{n} (x) | \leq x^{W_{n}} \exp (O_{x} (n)) (x > 1 fixed) .

Set $ξ = F (a / b)$ , $Q_{n} = b^{W_{n}} U_{n} (a / b)$ and $P_{n} = b^{W_{n}} V_{n} (a / b)$ . With

α = (C_{1} - C_{0}) \log a, τ = C_{0} \log a - C_{1} \log b > 0,

we have $\log (Q_{n} ξ - P_{n}) = - τ n^{2} + o (n^{2})$ and $| Q_{n} | \leq \exp (α n^{2} + o (n^{2}))$ .

For integers $A, B, p, q$ with $q > 0$ , if $L = A ξ - B$ and $2 q | L | \leq 1$ , then

| L | \leq | A | | ξ - p / q | .

When $A p - B q = 0$ this is equality. Otherwise the nonzero integer $A p - B q$ gives $1 / q \leq | L | + | A | | ξ - p / q |$ , which proves the inequality. Fix $0 < η < τ$ . For all sufficiently large $n$ , the estimates above give

e^{- (τ + η) n^{2}} \leq Q_{n} ξ - P_{n} \leq e^{- (τ - η) n^{2}}, | Q_{n} | \leq e^{(α + η) n^{2}} .

For every sufficiently large denominator $q$ , the upper remainder bound suggests choosing $n = ⌈ \sqrt{\log (2 q) / (τ - η)} ⌉$ . Then $2 q (Q_{n} ξ - P_{n}) \leq 1$ , so the preceding integer argument applies to $(A, B) = (Q_{n}, P_{n})$ for every numerator $p$ . Also $Q_{n} \neq 0$ : otherwise $Q_{n} ξ - P_{n}$ would be an integer strictly between $0$ and $1$ . The lower remainder bound and the coefficient bound now give

| ξ - p / q | \geq e^{- (α + τ + 2 η) n^{2}} = q^{- (α + τ + 2 η) / (τ - η) - o (1)},

since $n^{2} = \log (2 q) / (τ - η) + O_{η} (\sqrt{\log q} + 1)$ .

Let $η ↓ 0$ . The resulting bound is $1 + α / τ = (1 - θ) / (θ^{*} - θ)$ . Taking a common power multiplies $α, τ$ by $r$ , leaving this quotient unchanged; the constants in the approximation inequality may depend on $r$ . For $31 / 4$ , the exact rational interval calculation below gives

\begin{matrix} 0.40568302137302 < θ^{*} < 0.40568302139506, \\ 0.40369817316419 < \frac{\log 4}{\log 31} < 0.40369817316420 . \end{matrix}

For each trigamma difference, sum $k = 0, \dots, 255$ exactly and bound the remaining decreasing positive summand $f_{u, v} (x) = (x + u)^{- 2} - (x + v)^{- 2}$ by its integral from $256$ to infinity and that integral plus $f_{u, v} (256)$ . Bound $π$ with Machin’s identity and alternating arctangent series, and logarithms with the positive $arctanh$ series and a geometric tail bound. Outward rational interval operations then place $(1 - ρ) / (θ^{*} - ρ)$ , where $ρ = \log 4 / \log 31$ , between $300.4269130$ and $300.4269164$ , hence below $301$ . All endpoints are rational. For the bound $301$ alone, the coarser inequalities $ρ < 0.4036982$ and $θ^{*} > 0.40568$ suffice:

\frac{1 - ρ}{θ^{*} - ρ} < \frac{1 - 0.4036982}{0.40568 - 0.4036982} = \frac{2981509}{9909} < 301.

Here the quotient is increasing in $ρ$ and decreasing in $θ^{*}$ because $ρ < θ^{*} < 1$ . The finer enclosure also certifies the displayed decimal approximation. Lean checks the coarser calculation, including $2981509 / 9909 < 301$ , and proves enclosures of $θ^{*}$ and $ρ$ sharper than the finer ones displayed above (Section 2.5 and Theorem 2.3); the $256$ -summand calculation is an exact computation outside Lean. ◻

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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