Geometric Moments and Rational Lambert Values: Proofs and Further Results

Will Cook

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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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What a larger region would require.

At $3 / 2$ the logarithmic parameter is

\frac{\log 2}{\log 3} = 0.6309297535714574 \dots,

which exceeds $θ^{*}$ by $0.2252467 \dots$ . Theorem 2.2 gives no conclusion when $\log b / \log a \geq θ^{*}$ . The same argument can apply to another family once its polynomial integrality, degrees and fixed-base remainder estimates are proved. A smaller ratio of the resulting constants $C_{1} / C_{0}$ would enlarge the reciprocal-cutoff region. For example, $C_{1} / C_{0} \leq 2.4234$ would give a cutoff greater than $0.4126$ and admit $29 / 4$ . An integer-base irrationality-exponent bound alone does not supply the polynomial data needed for this specialisation. The existing formal proofs establish irrationality and measure bounds for the constructed forms; they do not supply a different family whose divided remainder is small and nonzero at $3 / 2$ .

The next restriction concerns one polynomial family whose estimates hold at every fixed $x > 1$ , with the same leading constants but possibly base-dependent error terms. This quantifier is essential: the proof chooses a large integer base at which remainder decay would outpace coefficient growth. Estimates only at $3 / 2$ , or for a different family at each base, do not permit that choice. The 2004 family satisfies the hypotheses. The conclusion concerns forms in $F (x)$ and $1$ , not simultaneous forms in several independent target values. Here height means the largest absolute coefficient. To compare it with the sum of the absolute coefficients, note that for a polynomial of degree at most $d$ ,

max_{j} | p_{j} | \leq \sum_{j} | p_{j} | \leq (d + 1) max_{j} | p_{j} | .

For a nonzero polynomial of degree $d = O (n^{2})$ , the logarithms of these norms differ by $O (\log n)$ . They have the same leading quadratic growth rate, but the literal zero-height conditions are not identical: $1 + X$ has maximum coefficient $1$ and coefficient sum $2$ . In particular, when $h = 0$ , the displayed bound $h n^{2} (1 + o (1))$ is zero, not an arbitrary $o (n^{2})$ term. The proof below also works with the additive bound $h n^{2} + o (n^{2})$ ; its evaluation estimate absorbs the $O (\log n)$ difference. Alternatively, either norm convention implies the hypothesis for the other norm after replacing $h$ by any larger positive constant. The degree conclusion is independent of that replacement. In the rational-base conclusions, write $a / b$ with integers $a > b \geq 1$ . The estimates hold without coprimality; a reduced representation gives the smaller denominator-clearing factor.

Theorem 2.5 (a degree restriction for estimates valid at every base). Let $(U_{n}, V_{n}) \in Z [x]^{2}$ be a sequence such that, for constants $σ, δ > 0$ and $h \geq 0$ independent of $n$ and of the base,

$Λ_{n} (x) := U_{n} (x) F (x) - V_{n} (x) \neq 0$ for every real $x > 1$ ;
$\deg U_{n}, \deg V_{n} \leq δ n^{2} (1 + o (1))$ ;
the coefficient heights satisfy
$\log max (H (U_{n}), H (V_{n})) \leq h n^{2} (1 + o (1)),$
where $H (P)$ is the largest absolute value of a coefficient of $P$ ;
the remainders satisfy
$\log | Λ_{n} (x) | = - σ n^{2} \log x (1 + o (1))$
for every real $x > 1$ .

Put $d_{n} := max (\deg U_{n}, \deg V_{n})$ . Then $σ \leq δ$ , and for every fixed rational base $a / b > 1$ ,

\underset{n \to \infty}{lim sup} n^{- 2} \log | b^{d_{n}} Λ_{n} (a / b) | \leq δ \log b - σ \log (a / b) .

Consequently the homogenised forms tend to zero whenever $\log b / \log a < σ / (σ + δ)$ , a sufficient region whose cutoff is at most $1 / 2$ . If the actual degrees satisfy $d_{n} / n^{2} \to d$ , then $d \geq σ$ and the limit exists and equals $d \log b - σ \log (a / b)$ ; in that case the forms tend to zero below $\log b / \log a = σ / (σ + d)$ and their absolute values tend to infinity above it, so the exact-degree case has no decaying homogenised forms at $3 / 2$ .

Proof. The coefficient-height bound controls evaluation at an integer base. For every fixed $p > 1$ ,

| U_{n} (p) |, | V_{n} (p) | \leq (d_{n} + 1) max (H (U_{n}), H (V_{n})) p^{d_{n}} \leq \exp ((h + δ \log p) n^{2} + o (n^{2})) .

Here $\log (d_{n} + 1) = o (n^{2})$ by the degree bound. This is where the coefficient-height hypothesis is needed.

Suppose $σ > δ$ . To make remainder decay beat coefficient growth, choose an integer $p \geq 2$ with $(σ - δ) \log p > h$ and put $u_{n} = U_{n} (p)$ , $v_{n} = V_{n} (p)$ and $ℓ_{n} = u_{n} F (p) - v_{n}$ . The integer determinant eliminates $F (p)$ ; the evaluation bound and hypothesis (4) give

u_{n} v_{n + 1} - u_{n + 1} v_{n} = u_{n + 1} ℓ_{n} - u_{n} ℓ_{n + 1} = o (1) .

Indeed, each product on the right has absolute value at most $\exp ((h + (δ - σ) \log p) n^{2} + o (n^{2}))$ . The expression on the left is an integer, so it is eventually zero. Also $u_{n} \neq 0$ eventually, since $u_{n} = 0$ would make $ℓ_{n} = - v_{n}$ a nonzero integer of absolute value less than $1$ . Thus $v_{n} / u_{n}$ is eventually a fixed rational number $r$ . If $F (p) = r$ , then $ℓ_{n} = 0$ ; otherwise $| ℓ_{n} | = | u_{n} | | F (p) - r | \geq | F (p) - r |$ . Both contradict the nonzero remainders tending to zero. Hence $σ \leq δ$ .

For a fixed rational base $a / b > 1$ , taking logarithms gives

n^{- 2} \log | b^{d_{n}} Λ_{n} (a / b) | = \frac{d_{n}}{n^{2}} \log b - σ \log (a / b) + o (1) .

Hypothesis (2) yields the stated upper limit and sufficient region. Since $σ \leq δ$ , its cutoff is at most $1 / 2$ . If $d_{n} / n^{2} \to d$ , apply the preceding integer-base argument with $d + ε$ in place of $δ$ for every $ε > 0$ to obtain $d \geq σ$ . The displayed identity then gives the exact limit. Its sign is that of $(σ + d) \log b - σ \log a$ , which proves both assertions away from equality. At $3 / 2$ this sign is positive because $\log 2 / \log 3 > 1 / 2 \geq σ / (σ + d)$ . ◻

Corollary 2.6 (nondecay when $b < a < b^{2}$ ). Under the hypotheses of Theorem 2.5, for positive integers $a, b$ with $b < a < b^{2}$ , the undivided forms $b^{d_{n}} Λ_{n} (a / b)$ do not tend to zero. No limit of $d_{n} / n^{2}$ is assumed.

Proof. Write $x = a / b$ and suppose $C_{n} = b^{d_{n}} Λ_{n} (x) \to 0$ . Since $C_{n} \neq 0$ and $| C_{n} | \leq 1$ eventually, logarithms and the remainder asymptotic at $x$ give the degree bound

d_{n} \log b \leq - \log | Λ_{n} (x) | = σ \log x n^{2} + o (n^{2}) .

Set $c = σ \log x / \log b$ . Since $1 < x < b$ , we have $0 < c < σ$ , and the displayed inequality gives $d_{n} \leq (c + ε) n^{2}$ eventually for each $ε > 0$ . Apply Theorem 2.5 to the same polynomial family with degree upper rate $c$ : its height bound, nonvanishing and all-base remainder asymptotics are unchanged. This gives $σ \leq c$ , contradicting $c < σ$ . ◻

This strengthens the failure of a sufficient-cutoff test to an exclusion of decay under the stated all-base hypotheses. It does not supply an actual-degree limit or assert divergence. If $d_{n} / n^{2} \to d$ , the separate exact-degree result gives the stronger conclusion $| b^{d_{n}} Λ_{n} (a / b) | \to \infty$ in the same strict region. Equality $a = b^{2}$ remains unclassified. The corollary concerns the undivided forms; it does not exclude base-dependent content division or other irrationality methods outside its hypotheses.

A degree bound alone would not supply the evaluation estimate: even degree-zero polynomials can have arbitrarily large integer coefficients. This explains the height hypothesis (3), without establishing that it cannot be weakened or omitted. At $\log b / \log a = σ / (σ + d)$ , the normalised logarithm is zero and the hypotheses decide neither behaviour. Theorem 2.5 is a restriction on the stated families, not on all Padé constructions.

There is also a distinction between the degree cutoff and an irrationality-exponent estimate at a fixed integer base $p \geq 2$ . The hypotheses above give

| U_{n} (p) | \leq \exp ((h + δ \log p) n^{2} + o (n^{2})), | Λ_{n} (p) | = \exp (- σ \log p n^{2} + o (n^{2})) .

Apply the integer-form argument in the proof of Corollary 2.4, with coefficient growth rate $h + δ \log p$ and remainder decay rate $σ \log p$ . It gives

μ_{i r r} (F (p)) \leq 1 + \frac{δ}{σ} + \frac{h}{σ \log p} .

Thus the reciprocal $σ / (σ + δ)$ of the degree expression $1 + δ / σ$ is not, under these hypotheses alone, the reciprocal of the exponent bound furnished by that argument. To obtain the latter identification it suffices to prove the stronger evaluated estimate $| U_{n} (p) | \leq \exp (δ \log p n^{2} + o (n^{2}))$ . This is separate from bounding the polynomial’s coefficients by $\exp (O (n^{2}))$ .

For the family of Section 2.1 the fourth hypothesis is the size estimate proved there and the second is (8). The third is proved next, so Theorem 2.5 applies to that family.

Lemma 2.7 (coefficient heights of the constructed polynomials). There is a constant $h$ with $\log max (H (U_{n}), H (V_{n})) \leq h n^{2}$ for every $n \geq 1$ , where $U_{n}$ and $V_{n}$ are the polynomials of (7).

Proof. For a polynomial or Laurent polynomial $P$ , write $‖ P ‖$ for the sum of the absolute values of its coefficients. Then $‖ P Q ‖ \leq ‖ P ‖ ‖ Q ‖$ ; for an ordinary polynomial, $H (P) \leq ‖ P ‖$ . This convention includes the Laurent monomials in $c_{n, k}$ .

By Lemma 2.1 every $ν_{n, l}$ is $0$ or $1$ , so $D_{N} / Ω_{n} = \prod_{l \in S} Φ_{l}$ over a subset $S \subseteq {1, \dots, N}$ . A monic polynomial with roots $ζ_{1}, \dots, ζ_{d}$ on the unit circle has coefficient norm at most $2^{d}$ : expanding its linear factors bounds the sum of the absolute coefficients by $\prod_{j = 1}^{d} (1 + | ζ_{j} |) = 2^{d}$ . Hence $‖ Φ_{l} ‖ \leq 2^{φ (l)}$ and

‖ \frac{D_{N}}{Ω_{n}} ‖ \leq 2^{\sum_{l \leq N} φ (l)} \leq 2^{225 n^{2}} .

For $A_{n}$ , the Gaussian binomial ${[\binom{m}{r}]}_{X}$ has nonnegative coefficients summing to $(\binom{m}{r})$ , so $‖ c_{n, k} ‖ \leq (\binom{k - 1}{a_{1} - 1}) (\binom{β_{n} - a_{2} - 1}{β_{n} - k - 1}) \leq 2^{2 β_{n}}$ by (3), and summing the at most $β_{n}$ terms of (4) gives $‖ A_{n} ‖ \leq β_{n} 2^{2 β_{n}}$ . Hence $‖ U_{n} ‖ \leq ‖ D_{N} / Ω_{n} ‖ ‖ A_{n} ‖ \leq e^{O (n^{2})}$ and $H (U_{n}) \leq e^{O (n^{2})}$ .

For $V_{n}$ , bound it on the circle $| z | = 2$ and use Cauchy’s estimate $H (V_{n}) \leq max_{| z | = 2} | V_{n} (z) |$ : each coefficient satisfies $| v_{i} | \leq 2^{- i} max_{| z | = 2} | V_{n} (z) |$ . The bound also holds for the zero polynomial. On that circle $| z^{l} - 1 | \geq 2^{l} - 1 \geq 1$ , so each of the at most $β_{n} + a_{0}$ inner terms of (5) has modulus at most $1$ ; also $| c_{n, k} (z) z^{a_{0} k} | \leq ‖ c_{n, k} ‖ 2^{K_{n}}$ and $| D_{N} (z) / Ω_{n} (z) | \leq ‖ D_{N} / Ω_{n} ‖ 2^{225 n^{2}} \leq e^{O (n^{2})}$ , while $| z^{- M_{n}} | \leq 1$ . There are $O (n)$ outer summands, each with $O (n)$ inner terms. After multiplication by the normalising factor, each term is bounded by $e^{O (n^{2})}$ , since $K_{n} = O (n^{2})$ . Summing the $O (n^{2})$ terms preserves this bound. Hence $max_{| z | = 2} | V_{n} (z) | \leq e^{O (n^{2})}$ and $H (V_{n}) \leq e^{O (n^{2})}$ . ◻

With Lemma 2.7, Zudilin’s family satisfies all four hypotheses of Theorem 2.5, with exact degree limit $d = C_{1} - C_{0}$ by (10), and decay exponent $σ = C_{0}$ because $\log Λ_{n} (x) = - (K_{n} - W_{n}) \log x + o (n^{2})$ for each fixed real $x > 1$ by the size estimate of Section 2.4. For it $σ / (σ + d) = θ^{*}$ and $(σ + d) / σ = μ$ . The stronger bound $| U_{n} (p) | \leq p^{W_{n}} \exp (O_{p} (n))$ from Corollary 2.4 removes the extra height term at each integer base. Within this family the rational-base threshold is therefore the reciprocal of the proved integer-base irrationality-exponent bound.

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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