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.