Hostname: page-component-7c8c6479df-24hb2 Total loading time: 0 Render date: 2024-03-19T05:20:37.914Z Has data issue: false hasContentIssue false

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
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)