Irrationality criteria for Lambert subseries

Cyclotomic prime-power fact.

Let $b\ge 2$, $n\ge 2$, $\ell \mid {\mathrm{\Phi }}_n(b)$ be prime and $e=v_{\ell }(b^n-1)$. Then ${\mathrm{ord}}_{{\ell }^e}(b)=n$. We include the $2$-adic case. For odd $\ell$, put $d={\mathrm{ord}}_{\ell }(b)$ and $s=v_{\ell }(b^d-1)$. The elementary lifting identity $v_{\ell }(b^{dt}-1)=s+v_{\ell }(t)$ follows by factoring a geometric sum when $\ell \nmid t$ and by a binomial expansion for a factor $\ell$. In the factorisation $b^n-1=\prod _{r\mid n}{\mathrm{\Phi }}_r(b)$, subtracting these valuations over proper divisors gives

If $n=d$, no smaller positive exponent gives $b^m\equiv 1(\mathrm{mod}\ell )$. If $n=d{\ell }^j$, the full valuation is $e=s+j$; the same lifting identity shows that the least exponent giving valuation $e$ is $d{\ell }^j=n$. For $\ell =2$, $b$ is odd. Factoring $b^{2t}-1$ gives

The only possible indices $n\ge 2$ are $n=2^j$: $v_2({\mathrm{\Phi }}_2(b))=v_2(b+1)$ and $v_2({\mathrm{\Phi }}_{2^j}(b))=1$ for $j\ge 2$; all other indices have valuation zero. The displayed identity again makes $2^j$ the least exponent with the full valuation. This proves the fact. Finally ${\mathrm{\Phi }}_n(b)>1$: each primitive $n$th root $\zeta$ satisfies $|b-\zeta |>1$, and their product is ${\mathrm{\Phi }}_n(b)$.

Exact scalar cover cost.

For $t\ge 1$, put $z={\log }_{2}t$. The substitution $y=2^{\alpha }$ reduces the infimum defining $\mathrm{\Psi }(t)$ to $y^z/(y-1)$ on $1<y\le 2$. Its derivative has the sign of $(z-1)y-z$. Hence

The minimum is at $\alpha =1$ for $t\le 4$ and in the interior for $t>4$; the quotient diverges as $\alpha \downarrow 0$. This scalar identity does not supply a converse to the positive-cover criterion.