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On Benford's law: The first digit problem

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Probability Theory and Mathematical Statistics

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References

  1. R. L. Adler and A.G. Konheim, Solution of Problem 4999, Amer. Math. Monthly 70 (1963), 218–219.

    Google Scholar 

  2. V. I. Arnol'd and A. Avez, Problèmes ergodiques de la mécanique classique, Gauthier-Villars, Paris 1967.

    MATH  Google Scholar 

  3. P. Billingsley, Ergodic theory and information, John Wiley and Sons, Inc., New York-London-Sydney 1965.

    MATH  Google Scholar 

  4. W. G. Brady, More on Benford's law, Fibonacci Quart. 16 (1978), 51–52.

    MATH  Google Scholar 

  5. J. L. Brown and R. L. Duncan, Modulo one uniform recurrence of the second order, Proc. Amer. Math. Soc. 50 (1975), 101–106.

    Article  MathSciNet  Google Scholar 

  6. J. G. van der Corput, Diophantische Ungleichungen, Acta Math. 56 (1931), 373–456.

    Article  MathSciNet  MATH  Google Scholar 

  7. P. Diaconis, The distribution of leading digits and uniform distribution mod 1, Ann. Prob. 5 (1977), 72–81.

    Article  MathSciNet  MATH  Google Scholar 

  8. R. L. Duncan, An application of uniform distributions to the Fibonacci numbers, Fibonacci Quart. 5 (1967), 137–140.

    MathSciNet  MATH  Google Scholar 

  9. R. L. Duncan, Note on the initial digit problem, ibid. 7 (1969), 474–475.

    MATH  Google Scholar 

  10. B. J. Flehinger, On the probability that a random integer has initial digit A, Amer. Math. Monthly 73 (1966), 1056–1061.

    Article  MathSciNet  MATH  Google Scholar 

  11. K. Goto and T. Kano, Uniform distribution of some special sequences, Proc. Japan Acad. Ser. A Math. Sci. 61 (1985), 83–86.

    Article  MathSciNet  MATH  Google Scholar 

  12. G. H. Hardy, Divergent series, Oxford UP, London 1949.

    Google Scholar 

  13. A. Ja. Hincin, Continued fractions, The University of Chicago Press, Chicago, Ill.-London 1964.

    Google Scholar 

  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. 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.

    MathSciNet  MATH  Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

  17. L. Kuipers and H. Niederreiter, Uniform distribution of sequences, John Wiley and Sons, Inc., New York-London-Sydney-Toronto 1974.

    MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  19. K. Nagasaka, On Benford's law, Ann. Inst. Stat. Math. 36 (1984), 337–352.

    Article  MathSciNet  MATH  Google Scholar 

  20. K. Nagasaka, Statistical properties of arithmetical sequences, Doctoral thesis, Tokyo Inst. Technology 1987.

    Google Scholar 

  21. K. Nagasaka and J.-S. Shiue, Benford's law for linear recurrence sequences, submitted to Tsukuba J. Math.

    Google Scholar 

  22. R. A. Raimi, The first digit problem, Amer. Math. Monthly 83 (1976), 521–538.

    Article  MathSciNet  MATH  Google Scholar 

  23. G. Rauzy, Propriétés statistiques de suites arithmétiques, Presses Univ. France, Paris 1976.

    MATH  Google Scholar 

  24. P. Schatte, Zur Verteilung der Mantissa der Gleichkommadarstellung einer Zerfallsgrösse, Z. Angew. Math. Mech. 53 (1973), 553–565.

    Article  MathSciNet  MATH  Google Scholar 

  25. P. Schatte, On H-summability and the uniform distribution of sequences, Math. Nachr. 113 (1983), 237–243.

    Article  MathSciNet  MATH  Google Scholar 

  26. P. Schatte, Estimates for the H-uniform distribution, preprint.

    Google Scholar 

  27. P. Schatte, On the uniform distribution of certain sequences and Benford's law, to appear in Math. Nachr.

    Google Scholar 

  28. L. C. Washington, Benford's law for Fibonacci and Lucas numbers, Fibonacci Quart. 19 (1981), 175–177.

    MathSciNet  MATH  Google Scholar 

  29. R.E. Whitney, Initial digits for the sequence of primes, Amer. Math. Monthly 79 (1972), 150–152.

    Article  MathSciNet  MATH  Google Scholar 

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Shinzo Watanabe Jurii Vasilievich Prokhorov

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© 1988 Springer-Verlag

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Kanemitsu, S., Nagasaka, K., Rauzy, G., Shiue, JS. (1988). On Benford's law: The first digit problem. In: Watanabe, S., Prokhorov, J.V. (eds) Probability Theory and Mathematical Statistics. Lecture Notes in Mathematics, vol 1299. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0078471

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  • DOI: https://doi.org/10.1007/BFb0078471

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  • Print ISBN: 978-3-540-18814-8

  • Online ISBN: 978-3-540-48187-4

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