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

TL;DR

This paper explores the relationship between the base $b$ expansion of the limit of normalized trailing zeroes in factorials and the actual zero counts at powers of $b$, revealing precise digit correspondences in prime power cases.

Contribution

It establishes a connection between the base $b$ expansion of $ heta(b)$ and the trailing zero counts of $b^k!$, with improved bounds for square-free bases.

Findings

Digits of $Z_b(b^k)$ match the first $k$ digits of $ heta(b)$'s fractional part for prime power bases.

The discrepancy in digit correspondence is bounded by $loor{ ext{log}_b(k)+3}$ in the general case.

Conjectures suggest this bound can be reduced for square-free bases.

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.