Journal of the London Mathematical Society 74, pp. 289-303.
ISSN/ISBN: 0024-6107 DOI: 10.1112/S0024610706023131
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)}(x_{0}) for 1≤k≤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≤k≤N} is small.
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},
}
Reference Type: Journal Article
Subject Area(s): Number Theory, Probability Theory