Let $\frac{a}{b}$ be a rational number strictly between $0$ and $1$ , and let: $$\frac{a}{b}=\sum_{k=1}^{N}\frac{1}{n_{k}}$$ be an Egyptian fraction representation of $\frac{a}{b}$ . In my current research, I very much need to have an explicit upper bound for $N$ (the “length” of the Egyptian fraction) in terms of the denominator, $b$ .
I have found several papers (Vose (1985), Tennenbaum & Yakoda (1990), and so on) on the topic. Vose's result (that $N\in O\left(\sqrt{\ln b}\right)$) is exactly what I need, save for one critical detail: I need to have an upper bound that isn't given as an asymptotic, but rather, as an inequality, one in which the pertinent constant $K$ such that: $$N\leq K\sqrt{\ln b}$$ are explicitly given (ex. $K=e^{e^{b}}$ , or whatever). Additionally, Tennenbaum & Yakoda point out that Vose's bound "holds uniformly for $N\geq3$", which (correct me if I'm wrong) I believe means that there is a $K$ which works for all $N\geq3$. Knowing that $K$ explicitly would be a godsend for my research.
Most of the results along these lines which I have found thus far tend to be insufficient for my purposes. On the other hand, the results that are of use to me seem to have constants which depend on quantities that are themselves asymptotics of other quantities, or—worse yet—are not explicitly constructed. Moreover, the paper at the heart of the matter (Erdös (1950)) only seems to be available in Hungarian, a language with which I have rather little skill (i.e., none whatsoever).
Any assistance would be greatly appreciated.