Geometric Moments and Rational Lambert Values: Proofs and Further Results

A limitation of the rectangular exponent model

We conclude the comparison of exponent models by bounding ${\mathrm{\Theta }}_{\mathrm{HP}}$, defined in Section 3. For $\rho \ge 0$ and $\sigma \ge 1+\rho$ its denominator

is positive. Put $u=\sigma -1-\rho \ge 0$. Multiplying the difference ${\mathrm{\Theta }}_{\mathrm{HP}}(\rho ,\sigma )-(1/2-1/{\pi }^2)$ by $2{\pi }^2$ times this denominator gives exactly

Every term is nonpositive. The original difference has the same sign, and it vanishes exactly when $\rho =0$ and $\sigma =1$ (exact expansion, nonpositivity, equality case). Equivalently, within that model the displayed threshold never exceeds the classical one-function margin, with equality only at the classical endpoint (bound, equality). This bounds only the displayed exponent model. Applying it to an approximation family would require the polynomial construction, integrality and asymptotic estimates connecting that family to ${\mathrm{\Theta }}_{\mathrm{HP}}$. The rational cutoff $81/200$ discussed above satisfies

For a sufficient criterion of the form $\log b/\log a<T$, including $3/2$ requires $T>\log 2/\log 3$, not merely an improvement on $81/200$. This numerical target applies to that form of criterion, not to every irrationality argument.

A concrete optimisation question is available within the 2004 construction itself. Write a source parameter direction as $\alpha =({\alpha }_0,{\alpha }_1,{\alpha }_2;\beta )$, with positive integer entries satisfying

The source parameters are $a_j={\alpha }_{j}n+1$ and $b=\beta n+2$; here $b$ is not a rational-base denominator. The gcd condition is a normalisation of the parameter direction, not a primitivity assertion about evaluated rows. Its invariance under dilation is checked below. For these directions set

and define the periodic step function

Zudilin’s constants in (25) and (26), already used in the finite direction scan, are

Here ${\psi }_1$ is the trigamma function used earlier. The step function is bounded, nonnegative, periodic with period $1$, and zero near zero, so the integral is finite.

The gcd normalisation does not change the ratio being optimised. Indeed, periodicity and $-{\psi }_1^{\prime }(u)=2\sum _{j\ge 0}(u+j{)}^{-3}$ give, by Tonelli’s theorem,

For any positive integer $k$, the floor formulas give ${\omega }_{k\alpha }(u)={\omega }_{\alpha }(ku)$. Substitution in the last integral therefore multiplies it by $k^2$. All other terms in $C_0$ and $C_1$ are quadratic in the direction, including $m^2$. Hence $C_i(k\alpha )=k^2{C}_i(\alpha )$ for $i=0,1$, so passing to a primitive direction preserves both positivity and $C_0/C_1$. Let $\mathcal{A}$ be exactly the displayed primitive directions with $C_0(\alpha )>0$ and $C_1(\alpha )>0$.

The immediate bounds are ${\theta }^{\ast }\le \underset{\mathcal{A}}{sup}{C}_0/C_1\le 1/2$. Indeed, at any integer base $p\ge 2$ the source gives ${\mu }_{\mathrm{i}\mathrm{r}\mathrm{r}}(F(p))\le C_1(\alpha )/C_0(\alpha )$, whereas pigeonhole approximation gives ${\mu }_{\mathrm{i}\mathrm{r}\mathrm{r}}(F(p))\ge 2$. Thus even the optimal reciprocal criterion in this class cannot exceed $1/2$, which is below $\log 2/\log 3$. Optimising these directions cannot include $3/2$ by that criterion. For other noninteger bases, a new direction would still require the polynomial degree and coefficient-height estimates used in the rational-base transfer; the integer-base source alone does not supply that application. The unrestricted question of which rational bases give irrational values is not settled here and is not reduced to any one of these problems.