Geometric Moments and Rational Lambert Values: Proofs and Further Results

A certified finite parameter search.

The parameter ratios $(14,12,14;27)$ are Zudilin’s [8]. They maximise $C_0/C_1$ in the following finite box: positive integer parameters $({\alpha }_0,{\alpha }_1,{\alpha }_2;\beta )$ with every entry at most $30$, greatest common divisor $1$, and

These are the source’s parameter inequalities [8], with a bound imposed for enumeration. The definitions of $C_0,C_1$ and their floor function are reproduced in Section 10. There are exactly $\mathrm{37,533}$ tuples. Exactly two attain the maximum: $(14,12,14;27)$ and $(15,12,13;26)$. Both give ${\theta }^{\ast }$, although they are distinct primitive directions.

The enumeration and comparisons are certified by rational intervals. The fractions $j/c$ with $1\le c\le 30$ partition $[0,1)$ into $278$ half-open intervals. Every floor function occurring in the search is constant on each interval, and the first interval contributes zero. On each remaining interval $[u,v)$, the integral is a sum of positive differences $(k+u{)}^{-2}-(k+v{)}^{-2}$, weighted by $0$ or $1$. With $M=256$, the tail lies between

This follows by comparing the positive decreasing summand with its integral. Outward rounding of rational numbers and Machin’s bounds for $\pi$ give, for each of the two maximisers,

Every other tuple has $C_0/C_1<0.40563943278333$. The intervals therefore separate these two candidates from all other tuples, without relying on floating-point ordering.

Their exact equality is not inferred from overlapping intervals. Write ${\omega }_A,{\omega }_B$ for their step functions in the order just listed, and let $J_A,J_B$ be their integrals against $d(-{\psi }_1)$ on $[0,1]$. The floor formulas give

For every positive integer $c$, telescoping the $c$ intervals and splitting the defining trigamma series into residue classes gives

Hence $J_B-J_A={\pi }^2/3$. Both directions have $m=15$ and $C_1=1091/2$, while the quadratic part of $C_0$ before the $3/{\pi }^2$ correction is $266$ for $A$ and $265$ for $B$. Its decrease by $1$ exactly cancels the increase $3(J_B-J_A)/{\pi }^2=1$. Thus $C_0$ also agrees.

A calculation compares every tuple of the box with interval bounds, and a second implementation repeats it with $512$ summands before the tail and a different enumeration order; both are in one program, with its recorded result. The calculation proves optimality only in the displayed finite box. No bound on a direction outside that box follows, and this calculation is not needed for the irrationality theorem.

The two-coordinate forms in $1$ and $F(p)$ should be distinguished from simultaneous approximation to two target values. Postelmans and Van Assche prove the $\mathbb{Q}$-linear independence of $1,{\zeta }_q(1),{\zeta }_q(2)$ for $q=1/p$ with integer $p\ge 2$ [30]. Their confluent multiple little $q$-Jacobi construction has two orthogonality conditions; its common integer normalisation and nonvanishing argument are in Section 6, especially (6.1)–(6.4). That theorem treats two target values and retains the integer inverse-base restriction. It supplies neither the coefficient pairs nor the all-base estimates assumed in Theorem 2.5; it is not an application at $3/2$.