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Journal of the London Mathematical Society 2006 74(2):289-303; doi:10.1112/S0024610706023131
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© The London Mathematical Society 2006

BENFORD'S LAW FOR THE 3x + 1 FUNCTION

Jeffrey C. Lagarias and K. Soundararajan

Department of Mathematics, The University of Michigan Ann Arbor, MI 48109-1043, USA lagarias{at}umich.edu
Department of Mathematics, Stanford University Stanford, CA 94305-2125, USA ksound{at}math.stanford.edu

Received 14 September 2005. Revision received 10 January 2006.

Benford's law (to base B) for an infinite sequence {xk : k ≥ 1} of positive quantities xk is the assertion that {logB xk : k ≥ 1} is uniformly distributed (mod 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 xk = T(k)(x0) for 1 ≤ k ≤ N of the 3x + 1 function, where N is fixed. It shows that for most initial values x0, such sequences approximately satisfy Benford's law, in the sense that the discrepancy of the finite sequence {logB xk : 1 ≤ k ≤ N} is small.


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