Geometric Moments and Rational Lambert Values: Proofs and Further Results

Finite calculations

The polynomial reconstruction below determines the forms for $n=1,2,3,4$. Together with the exact comparison of all directions in the preceding finite box, it provides small cases against which to check the general formulas. The two calculations are available, along with the other finite calculations, in the reproduction guide, and a single script reruns them and compares each with its recorded result. The historical high-precision evaluations were not rerun. None of these finite calculations proves a statement for all indices, and none is a Lean proof. The tests for $N\le 8$ and the rational interval certificate are described separately.

The public reproduction guide gives the command and dependency pin for the direction-search program. Run it with --bound 30 and the output path named there; adding --check compares the result without replacing the stored search results. The stored record specifies mpmath 1.3.0 at thirty decimal digits. Its exact enumeration count is $\mathrm{37,533}$, but its ordering by $C_0/C_1$ is numerical and has no interval certificate in that historical record. The separate rational certificate above verifies the maximisers in the stated finite box; the archived program itself still uses numerical comparisons.

An exact integer calculation (program, every coefficient) reconstructs the forms of Section 2.1 directly in $\mathbb{Z}[X]$ from the displayed source formulas. It checks division by ${\mathrm{\Omega }}_n$ and $X^{M_n}$ with zero remainder and computes all coefficients of $U_n,V_n$. The exact degrees are

In each case $\mathrm{deg}{V}_n=W_n-1$ and both leading coefficients are $(-1{)}^n$. Independently expanding $F(1/q)=\sum _{j\ge 1}\tau (j)q^j$ checks the vanishing coefficients below order $K_n-W_n$ in $U_n(1/q)F(1/q)-V_n(1/q)$ and its next twelve coefficients against the positive source expression. Exact rational evaluation at $31/4$, $3$ and $7/2$ also checks homogenisation; at the two noninteger bases, the power $b^{W_n-1}$ fails to clear $U_n(a/b)$, as predicted by the general leading-coefficient argument.

A separate historical numerical report evaluated the remainder identity at those three bases to relative accuracy below ${10}^{-39}$ at $n=3$. It reported positivity of ${\hat{\mathrm{\Lambda }}}_n$ and agreement with the bounds on $H_n$ in Section 2.4. These high-precision real evaluations were not reproduced by the exact polynomial calculation just described.

The historical constant evaluation used $\omega$, the thirteen intervals of Lemma 2.1 and forty-digit trigamma values. It reported

and $C_1/C_0=2.46497868357497\dots$, agreeing with the source’s printed precision [8]. The rational interval estimate stated earlier, rather than these rounded digits, supports the numerical inequality.

The main term $K_n\log b-(K_n-W_n)\log a$ at $31/4$ is negative for every $1\le n\le 400$, as verified by exact rational logarithm enclosures and exact totient sums (program, result). Dividing by $n^2$ gives, to three decimal places, $-58.478,-10.280,-4.297,-3.863$ at $n=1,10,100,400$. This finite sign check does not assert negativity at every index. The proved limit is $C_1\log 4-C_0\log 31=-3.718\dots$; the elementary convergence-rate bound is $O({\log }^2(n+2)/n)$. Indeed, summing the $O(y\log (2y))$ endpoint errors over the floor blocks with $k\le n$ gives $O(n{\log }^2(n+2))$; the remaining main-term tail is $O(1)$ since its summands are $O(n^2/k^3)$. The linear terms of $M_n$ add $O(n)$. The stronger rate $O(1/n)$ does not follow from these estimates. Only the limit enters Theorem 2.2.

For orientation, the all-rank order and leading-coefficient formulas of Theorem 3.4 give the following first seven values: