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BENFORD'S LAW FOR THE $3x+1$ FUNCTION

Published online by Cambridge University Press:  25 October 2006

JEFFREY C. LAGARIAS
Affiliation:
Department of Mathematics, The University of Michigan, Ann Arbor, MI 48109-1043, USAlagarias@umich.edu
K. SOUNDARARAJAN
Affiliation:
Department of Mathematics, Stanford University, Stanford, CA 94305-2125, USAksound@math.stanford.edu
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Abstract

Benford's law (to base $B$) for an infinite sequence $\{x_k: k \ge 1\}$ of positive quantities $x_k$ is the assertion that $\{ \log_B x_k : k \ge 1\}$ is uniformly distributed $(\bmod\ 1)$. The $3x+1$ function $T(n)$ is given by $T(n)=(3n+1)/{2}$ if $n$ is odd, and $T(n)= n/2$ if $n$ is even. This paper studies the initial iterates $x_k= T^{(k)}(x_0)$ for $1 \le k \le N$ of the $3x+1$ function, where $N$ is fixed. It shows that for most initial values $x_0$, such sequences approximately satisfy Benford's law, in the sense that the discrepancy of the finite sequence $\{\log_B x_k: 1 \le k \le N \}$ is small.

Type
Notes and Papers
Copyright
The London Mathematical Society 2006

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