On the base $b$ expansion of the number of trailing zeroes of $b^k!$

Abstract

Let us denote by $Z_{b}(n)$ the number of trailing zeroes in the base b expansion of $n!$. In this paper we study the connection between the expression of $\vartheta(b):=\lim_{n\to \infty}Z_{b}(n)/n$ 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 $\lfloor \log_{b}(k) +3\rfloor $ digits. We finally show that this bound can be improved if $b$ is square-free and present some conjectures about this bound.