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N! has the first digit property. (English) Zbl 0627.10007
The author shows that the sequence N! obeys Benford’s law, i.e., $$\Pr (j=p)=\log_{10}(1+1/p)$$ where j is the first significant digit of N! and $$1\leq p\leq 9$$. The principal result is: if $$F=\{N!\}_{N=1,2,...}$$ and $$F_ k$$ are those in F whose first digit is k, then $$\lim \#(F_ k<m)/\#(F<m)=\log (1+1/k).$$ The proof is to show log($$\sqrt{2\pi n} (n/e)^ n)$$ is uniformly distributed modulo 1 using the Weyl criterion, and that if $$\{\sqrt{2\pi n} (n/e)^ n\}$$ is Benford so is $$\{$$ n!$$\}$$. The article concludes with a brief report on preliminary work investigating whether $$\{a^ p\}_{p,prime}$$ is Benford under other than natural densities.
Reviewer: G.Lord

##### MSC:
 11A63 Radix representation; digital problems 11K06 General theory of distribution modulo $$1$$