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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

11A63 Radix representation; digital problems
11K06 General theory of distribution modulo \(1\)