I am reading the paper "Birthday paradox, coupon collectors, caching algorithms and self-organizing search" by Flajolet, Gardier and Thimonier:
http://algo.inria.fr/flajolet/Publications/FlGaTh92.pdf
Equation (37) on pg. 226 reads $$ E\{C_m\} = \int_0^{\infty}(1-\Theta(t)) dt, \quad \mathrm{where} \quad \Theta(t)=\prod_{i=1}^m (1-e^{-p_i t}). $$ Here $m$ is a positive integer, and for all $i\in\{1, \ldots, m\}$, $p_i=\frac{1}{H_m i}$, where $H_m$ is the $m$-th harmonic number, $\sum_{k=1}^m \frac{1}{k}$.
The authors write
It can be proved that $\Theta(t)$ has a sharp transition from $0$ to $1$ for $t$ around $m \log^2 m$. More precisely, quantity $F_m(x)=-\log \Theta(xm \log m H_m)$ is such that for fixed $x$ as $m \to \infty$, we have: $F_m(x) \to \infty$ if $x<1$ and $F_m(x) \to 0$ if $x \geq 1$.
I don't understand this. Surely if $x>0$, then whichever side of $1$ $x$ is, $xm \log mH_m$ will tend to infinity as $m$ tends to infinity, hence $\Theta$ of it will tend to $1$ as $m$ tends to infinity, hence $F_m(x)$ will tend to $0$ as $m$ tends to infinity ?
For $F_m(x)$ to tend to infinity, wouldn't we need $\Theta$ to tend to $0$, for which, wouldn't we need $xm \log mH_m$ to tend to $0$ ?
Unless I've made a mistake, I suppose there must be some typos in the paper, but I can't immediately see what they might be.