BENFORD'S LAW FOR THE 3x + 1 FUNCTION |
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| Authors: | Lagarias, Jeffrey C. Soundararajan, K. |
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| Affiliation: | 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 |
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| Abstract: | 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 functionT(n) is given by T(n) = (3n + 1)/2 if n is odd, and T(n) = n/2if 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 showsthat for most initial values x0, such sequences approximatelysatisfy Benford's law, in the sense that the discrepancy ofthe finite sequence {logB xk : 1 k N} is small. |
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