Lagarias, JC and Soundararajan, K (2006)

Benford's law for the 3x+1 function

Journal of the London Mathematical Society 74, pp. 289-303.

ISSN/ISBN: 0024-6107 DOI: 10.1112/S0024610706023131

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• Zentralblatt: Zbl 1117.11018
• MathSciNet: MR2269630 (2007h:37007)
• For online information, click here.

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Abstract: Benford's law (to base B) for an infinite sequence {x_k: k≥1} of positive quantities x_k is the assertion that {log_B x_k : 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 x_k= 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 {log_B x_k: 1≤k≤N} is small.

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Bibtex:
@article {MR2269630, AUTHOR = {Lagarias, Jeffrey C. and Soundararajan, K.}, TITLE = {Benford's law for the {$3x+1$} function}, JOURNAL = {J. London Math. Soc. (2)}, FJOURNAL = {Journal of the London Mathematical Society. Second Series}, VOLUME = {74}, YEAR = {2006}, NUMBER = {2}, PAGES = {289--303}, ISSN = {0024-6107}, CODEN = {JLMSAK}, MRCLASS = {37A45}, MRNUMBER = {2269630 (2007h:37007)}, MRREVIEWER = {Radhakrishnan Nair}, DOI = {10.1112/S0024610706023131}, URL = {http://dx.doi.org/10.1112/S0024610706023131}, }

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Reference Type: Journal Article

Subject Area(s): Number Theory, Probability Theory

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