Can it be shown that a positive fraction of the base-$b$ digits of n! are nonzero (in the limit as $n\to\infty$)?



Nonzero digits in n!
The bound given by F. Luca and cited in Gjergji Zaimi's answer has been recently improved, although only by a factor $\log \log \log n$. Precisely, in *C. Sanna, [On the sum of digits of the factorial](http://dx.doi.org/10.1016/j.jnt.2014.09.003), J. Number Theory 147 (2015), 836--841*, the author proved that
$$\min\{s\_b(n!),s\_b(\Lambda\_n)\} > c\_b \log n \log \log \log n , $$
for all integers $n > e^e$, where $s\_b(\cdot)$ is the sum of base-$b$ digits function, $c\_b > 0$ is a constant depending only on $b$, and $\Lambda\_n$ is the least common multiple of $1, \ldots, n$. Clearly, this inequality also holds with $s\_b$ replaced by $t\_b$, i.e., the function counting the number of non-zero base-$b$ digits of its argument, since $t\_b(m) \geq s\_b(m)/b$ for each positive integer $m$.


