Acta Mathematica Hungarica

, Volume 139, Issue 1–2, pp 49–63 | Cite as

On the mantissa distribution of powers of natural and prime numbers

  • Shalom Eliahou
  • Bruno Massé
  • Dominique Schneider


Given a fixed integer exponent r≧1, the mantissa sequences of (n r ) n and of \({(p_{n}^{r})}_{n}\), where p n denotes the nth prime number, are known not to admit any distribution with respect to the natural density. In this paper however, we show that, when r goes to infinity, these mantissa sequences tend to be distributed following Benford’s law in an appropriate sense, and we provide convergence speed estimates. In contrast, with respect to the log-density and the loglog-density, it is known that the mantissa sequences of (n r ) n and of \({(p_{n}^{r})}_{n}\) are distributed following Benford’s law. Here again, we provide previously unavailable convergence speed estimates for these phenomena. Our main tool is the Erdős–Turán inequality.

Key words and phrases

Benford’s law mantissa prime number 

Mathematics Subject Classification

60B10 11B05 11K99 


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  1. [1]
    D. I. Cohen and T. M. Katz, Prime numbers and the first digit phenomenon, J. Number Theory, 18 (1984), 261–268. MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    P. Diaconis, The distribution of leading digits and uniform distribution mod 1, Ann. Probab., 5 (1977), 72–81. MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    M. Drmota and R. F. Tichy, Sequences, Discrepancies and Applications, Lecture Notes in Math. 1651, Springer-Verlag (Berlin, 1997). zbMATHGoogle Scholar
  4. [4]
    R. L. Duncan, Note on the initial digit problem, Fibonacci Quart., 7 (1969), 474–475. zbMATHGoogle Scholar
  5. [5]
    A. Fuchs and G. Letta, Le problème du premier chiffre décimal pour les nombres premiers, (French) [The first digit problem for primes], Electron. J. Combin., 3 (1996), R25. MathSciNetGoogle Scholar
  6. [6]
    D. E. Knuth, The Art of Computer Programming, Volume 2, Addison-Wesley (Reading, Massachusetts, 1969). Google Scholar
  7. [7]
    L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, Dover Publications (New York, 2006). Google Scholar
  8. [8]
    B. Massé and D. Schneider, On weighted densities and their connection with the first digit phenomenon, Rocky Mountain J. Math., 51 (2011), 1395–1415. CrossRefGoogle Scholar
  9. [9]
    P. N. Posch, A survey of sequences and distribution functions satisfying the first-digit-law, J. Stat. and Management Systems, 11 (2008), 1–19. MathSciNetzbMATHGoogle Scholar
  10. [10]
    R. Raimi, The first digit problem, Amer. Math. Monthly, 83 (1976), 521–538. MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    P. Ribenboim, The Little Book of Bigger Primes, 2nd ed., Springer (Berlin, 2004), 185 p. zbMATHGoogle Scholar
  12. [12]
    J. Rivat and G. Tenenbaum, Inégalités d’Erdős–Turán, Ramanujan J., 9 (2005), 111–121. MathSciNetzbMATHCrossRefGoogle Scholar
  13. [13]
    E. C. Titchmarsh, The Theory of the Riemann Zeta-function, 2nd ed., The Clarendon Press, Oxford University Press (New York, 1986). zbMATHGoogle Scholar
  14. [14]
    R. E. Whitney, Initial digits for the sequence of primes, Amer. Math. Monthly, 79 (1972), 150–152. MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2012

Authors and Affiliations

  • Shalom Eliahou
    • 1
    • 2
    • 3
  • Bruno Massé
    • 1
    • 2
    • 3
  • Dominique Schneider
    • 1
    • 2
    • 3
  1. 1.ULCOLMPA J. LiouvilleCalaisFrance
  2. 2.Univ. Lille Nord de FranceLilleFrance
  3. 3.CNRSParisFrance

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