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\begin{abstract}

Let $Z_{b}(n)$ denote the number of trailing zeroes in the base-$b$
expansion of $n!$. In this paper we study the connection between
the expression of $\vartheta(b):=\lim_{n\rightarrow \infty}Z_{b}(n)/n$ in 
base $b$, and that of $Z_{b}(b^{k})$.

In particular, if $b$ is a prime power, we will show the equality
between the $k$ digits of $Z_{b}(b^{k})$ and the first $k$ digits in
the fractional part of $\vartheta (b)$. In the general case we will
see that this equality still holds except for, at most, the last
$\left\lfloor \log _{b}(k)\ +3\right\rfloor $ digits. We finally
show that this bound can be improved if $b$ is square-free and
present some conjectures about this bound.
\end{abstract}




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