Given a positive rational $\alpha$ and a natural number $k$, let $N_k(\alpha)$ be the number of Egyptian fraction representations of $\alpha$ with smallest element ${1\over k}$ (see e.g. the survey "Paul Erdős and Egyptian Fractions" by Graham). For $s>0$ let $$S_s(\alpha):=\sum_{k\in\mathbb{N}}{N_k(\alpha)\over k^s}$$ and let $$t(\alpha)=\inf\{s: S_s(\alpha)<\infty\}.$$
My first question is what the above "threshold" value is, in at least one particular natural case:
Q1: What is $t(1)$? Or, failing that, can we compute any concrete examples of $t(\alpha)$?
Tentatively, I suspect that $t(\alpha)=1$ for every positive rational $\alpha$, but I don't see how to prove that. Basically, I need an appropriate bound on the number of Egyptian fraction representations of a given rational with a given smallest term, but I don't see how to get a bound which is good enough for this question.
My second question is about what happens, specifically, at $s=2$:
Q2: What is $S_2(1)$?