On the base b expansion of the number of trailing zeroes of bᵏ!

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Let us denote by Zb(n) the number of trailing zeroes in the base b expansion of n!. In this paper we study the connection between the expression of ϑ(b):=ₙ→ ∞Zb(n)/n in base b, and that of Zb(bᵏ). In particular, if b is a prime power, we will show the equality between the k digits of Zb(bᵏ) and the first k digits in the fractional part of ϑ (b). In the general case we will see that this equality still holds except for, at most, the last (k) +3 digits. We finally show that this bound can be improved if b is square-free and present some conjectures about this bound.