On Benford's law: The first digit problem

  • Shigeru Kanemitsu
  • Kenji Nagasaka
  • Gérard Rauzy
  • Jau-Shyong Shiue
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1299)


Maximum Modulus Fibonacci Number Summation Method Integer Sequence Linear Recurrence 
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  1. [1]
    R. L. Adler and A.G. Konheim, Solution of Problem 4999, Amer. Math. Monthly 70 (1963), 218–219.Google Scholar
  2. [2]
    V. I. Arnol'd and A. Avez, Problèmes ergodiques de la mécanique classique, Gauthier-Villars, Paris 1967.zbMATHGoogle Scholar
  3. [3]
    P. Billingsley, Ergodic theory and information, John Wiley and Sons, Inc., New York-London-Sydney 1965.zbMATHGoogle Scholar
  4. [4]
    W. G. Brady, More on Benford's law, Fibonacci Quart. 16 (1978), 51–52.zbMATHGoogle Scholar
  5. [5]
    J. L. Brown and R. L. Duncan, Modulo one uniform recurrence of the second order, Proc. Amer. Math. Soc. 50 (1975), 101–106.MathSciNetCrossRefGoogle Scholar
  6. [6]
    J. G. van der Corput, Diophantische Ungleichungen, Acta Math. 56 (1931), 373–456.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    P. Diaconis, The distribution of leading digits and uniform distribution mod 1, Ann. Prob. 5 (1977), 72–81.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    R. L. Duncan, An application of uniform distributions to the Fibonacci numbers, Fibonacci Quart. 5 (1967), 137–140.MathSciNetzbMATHGoogle Scholar
  9. [9]
    R. L. Duncan, Note on the initial digit problem, ibid. 7 (1969), 474–475.zbMATHGoogle Scholar
  10. [10]
    B. J. Flehinger, On the probability that a random integer has initial digit A, Amer. Math. Monthly 73 (1966), 1056–1061.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    K. Goto and T. Kano, Uniform distribution of some special sequences, Proc. Japan Acad. Ser. A Math. Sci. 61 (1985), 83–86.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    G. H. Hardy, Divergent series, Oxford UP, London 1949.Google Scholar
  13. [13]
    A. Ja. Hincin, Continued fractions, The University of Chicago Press, Chicago, Ill.-London 1964.Google Scholar
  14. [14]
    P. Kiss and R. Tichy, Distribution of the ratios of the terms of a second order linear recurrence, to appear in Indag. Math.Google Scholar
  15. [15]
    L. Kuipers, Remark on a paper by R. L. Duncan concerning uniform distribution mod 1 of the sequence of logarithms of Fibonacci numbers, Fibonacci Quart. 7 (1969), 465–466, 473.MathSciNetzbMATHGoogle Scholar
  16. [16]
    L. Kuipers and J.-S. Shiue, Remark on a paper by Duncan and Brown on the sequence of logarithms of certain recursive sequences, ibid. 11 (1973), 292–294.MathSciNetzbMATHGoogle Scholar
  17. [17]
    L. Kuipers and H. Niederreiter, Uniform distribution of sequences, John Wiley and Sons, Inc., New York-London-Sydney-Toronto 1974.zbMATHGoogle Scholar
  18. [18]
    L. Murata, On certain densities of sets of primes, Proc. Japan Acad. Ser. A Math. Sci. 56 (1980), 351–353; On some fundamental relatives among certain asymptotic densities, Math. Rep. Toyama Univ. 4 (1981), 47–61.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    K. Nagasaka, On Benford's law, Ann. Inst. Stat. Math. 36 (1984), 337–352.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    K. Nagasaka, Statistical properties of arithmetical sequences, Doctoral thesis, Tokyo Inst. Technology 1987.Google Scholar
  21. [21]
    K. Nagasaka and J.-S. Shiue, Benford's law for linear recurrence sequences, submitted to Tsukuba J. Math.Google Scholar
  22. [22]
    R. A. Raimi, The first digit problem, Amer. Math. Monthly 83 (1976), 521–538.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    G. Rauzy, Propriétés statistiques de suites arithmétiques, Presses Univ. France, Paris 1976.zbMATHGoogle Scholar
  24. [24]
    P. Schatte, Zur Verteilung der Mantissa der Gleichkommadarstellung einer Zerfallsgrösse, Z. Angew. Math. Mech. 53 (1973), 553–565.MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    P. Schatte, On H-summability and the uniform distribution of sequences, Math. Nachr. 113 (1983), 237–243.MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    P. Schatte, Estimates for the H-uniform distribution, preprint.Google Scholar
  27. [27]
    P. Schatte, On the uniform distribution of certain sequences and Benford's law, to appear in Math. Nachr.Google Scholar
  28. [28]
    L. C. Washington, Benford's law for Fibonacci and Lucas numbers, Fibonacci Quart. 19 (1981), 175–177.MathSciNetzbMATHGoogle Scholar
  29. [29]
    R.E. Whitney, Initial digits for the sequence of primes, Amer. Math. Monthly 79 (1972), 150–152.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • Shigeru Kanemitsu
    • 1
  • Kenji Nagasaka
    • 2
  • Gérard Rauzy
    • 3
  • Jau-Shyong Shiue
    • 4
  1. 1.Department of Mathematics Faculty of ScienceKyushu UniversityFukuokaJapan
  2. 2.Department of Mathematics Faculty of EducationShinshu UniversityNaganoJapan
  3. 3.Faculté des Sciences de Luminy Mathématique-InformatiqueMarseille Cédex 9France
  4. 4.Department of Mathematical SciencesUniversity of Nevada, Las VegasLas VegasUSA

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