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

v a r t h e t a (b) : = l i m_{n o i n f t y} 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

v a r t h e t a (b)

. In the general case we will see that this equality still holds except for, at most, the last

l f l o o r l o g_{b} (k) + 3 f l o o r

digits. We finally show that this bound can be improved if

b

is square-free and present some conjectures about this bound.

Full work available at URL: https://arxiv.org/abs/1002.4175

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