Geometric Moments and Rational Lambert Values: Proofs and Further Results

Examples of rational bases.

Among reduced $a/b$ with $1\le b<a\le 60$, the region of Theorem 2.2 has $137$ members, of which $78$ are non-integral. The two largest admitted logarithmic ratios belong to $53/5$ and $31/4$, respectively $0.000313$ and $0.001985$ below ${\theta }^{\ast }$; the smallest excluded ratio is that of $52/5$, namely $0.4073243836\dots$, which exceeds the cutoff by $0.001641$. A calculation with rational bounds for logarithms and for the cutoff checks all $1101$ reduced fractions in the stated range, these rounded margins and the four fixed-denominator comparisons below (program, result). The bases in this region and outside the region of [6] are those in the strip $s^{\mu }<r\le s^{{\mu }_{\mathrm{BV}}}$, which is empty for $s=2$ and $s=3$, is $\{31\}$ for $s=4$, and is $\{53,54,56\}$ for $s=5$. The strip is infinite: its width $s^{{\mu }_{\mathrm{BV}}}-s^{\mu }$ eventually exceeds $s$, so for every large enough denominator it contains an integer congruent to $1$ modulo $s$. Hence $31/4$ is the member of the strip with least denominator and least numerator.