Irrationality criteria for Lambert subseries

Finite-support denominator periods

The least common multiple of the selected exponents survives cancellation as the multiplicative order of the base modulo the reduced denominator.

Put $L=\mathrm{lcm}(F)$. Clearing denominators gives $D_F\mid b^L-1$, so the upper divisibility for the order is immediate. For the reverse, fix a divisibility-maximal $n\in F$ with $n\ge 2$ and choose a prime $\ell \mid {\mathrm{\Phi }}_n(b)$. With $e=v_{\ell }(b^n-1)$, the cyclotomic fact in Section E gives ${\mathrm{ord}}_{{\ell }^e}(b)=n$ for the full prime power ${\ell }^e$. Every other selected exponent $m$ has $n\nmid m$, hence $v_{\ell }(b^m-1)<e$. The $n$th summand has uniquely smallest $\ell$-adic valuation and cannot cancel. Thus ${\ell }^e\mid D_F$ and $n\mid {\mathrm{ord}}_{D_F}(b)$. Taking all maximal selected exponents proves the order statement; the size bound follows from ${\mathrm{ord}}_{D_F}(b)\mid \phi (D_F)<D_F$ when $L\ge 2$. The case $F=\{1\}$ has order one directly.

The unique-valuation argument also permits signs $\pm 1$ on the finite summands. This signed extension is not included in the formal statement. The distinction between primes and prime powers is visible in

In both examples $2$ has multiplicative order six modulo the denominator. The first combines orders two and three; the second retains the order-six prime power $9$, although no prime divisor of $63$ has order six. The boundary $F=\{1\}$ at base two has $D_F=L=1$. These lower denominator bounds give no upper height control for infinite partial sums and do not decide an infinite-support value. Van Assche instead constructs approximants for the full Lambert series and proves both nonvanishing and decay of the associated integer linear forms [14]. That approximation mechanism does not follow from denominator survival alone.

In particular, $1/21$ is not a finite subseries sum. If it were, the order of $2$ modulo $21$ would force every selected exponent to divide $6$. The exponent $1$ is excluded since its weight is $1$. Any sum containing $2$ or $3$ exceeds $1/21$, whereas the only nonzero remaining sum is $1/63$, from the exponent $6$. This will distinguish finite and infinite representations in the next appendix.