Research in Number Theory

, 1:22 | Cite as

Leading digit laws on linear Lie groups

Open Access


We study the leading digit laws for the matrix entries of a linear Lie group G. For non-compact G, these laws generalize the following observations: (1) the normalized Haar measure of the Lie group \(\mathbb {R}^{+}\) is dx/x and (2) the scale invariance of dx/x implies the distribution of the digits follow Benford’s law. Viewing this scale invariance as left invariance of Haar measure, we see either Benford or power law behavior in the significands from one matrix entry of various such G. When G is compact, the leading digit laws we obtain come as a consequence of digit laws for a fixed number of components of a unit sphere. The sequence of digit laws for the unit sphere exhibits periodic behavior as the dimension tends to infinity.


Benford’s law Digit laws Haar measure Matrix groups 

Mathematics subject classification

11K06 60F99 (primary) 28C10 15B52 15B99 (secondary) 


  1. 1.
    Anderson, T, Rolen, L, Stoehr, R: Benford’s Law for coefficients of modular forms and partition functions. Proc. Am. Math. Soc. 139(5), 1533–1541 (2011).MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Abramowitz, M, Stegun, I: Handbook of mathematical functions. Natl. Bur. Stan. Appl. Math. Ser. 55, 257 (1964).MathSciNetGoogle Scholar
  3. 3.
    Barrett, O, Firk, FWK, Miller, SJ, Turnage-Butterbaugh, C: From quantum systems to L-functions: pair correlation statistics and beyond, to appear in open problems in mathematics (Nash, JJR, Rassias, MTH, eds.)Springer-Verlag.Google Scholar
  4. 4.
    Becker, T, Corcoran, TC, Greaves-Tunnell, A, Iafrate, JR, Jing, J, Miller, SJ, Porfilio, JD, Ronan, R, Samranvedhya, J, Strauch, F: Benford’s law and continuous dependent random variables. arXiv version. [preprint 2015].
  5. 5.
    Benford, F: The law of anomalous numbers. Proc. Am. Philos. Soc. 78, 551–572 (1938).Google Scholar
  6. 6.
    Berger, A, Hill, TP: Benford online bibliography.
  7. 7.
    Berger, A, Hill, TP: A basic theory of Benford’s law. Probab. Surv. 8, 1–126 (2011).MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Berger, A, Hill, TP: An introduction to Benford’s law. Princeton University Press, Princeton, NJ (2015).CrossRefGoogle Scholar
  9. 9.
    Brown, J, Duncan, R: Modulo one uniform distribution of the sequence of logarithms of certain recursive sequences. Fibonacci Q. 8, 482–486 (1970).MATHMathSciNetGoogle Scholar
  10. 10.
    Conrey, JB, Farmer, D, Keating, P, Rubinstein, M, Snaith, N: Integral moments of L-functions. Proc. London Math. Soc. (3). 91(1), 33–104 (2005).MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Diaconis, P: The distribution of leading digits and uniform distribution mod 1. Ann. Probab. 5, 72–81 (1979).MathSciNetCrossRefGoogle Scholar
  12. 12.
    Erdős, L, Ramirez, JA, Schlein, B, Yau, H-T: Bulk Universality for Wigner Matrices. Comm. Pure Appl Math. 63(70), 895–925 (2010).MathSciNetGoogle Scholar
  13. 13.
    Erdős, L, Schlein, B, Yau, H-T: Wegner estimate and level repulsion for Wigner random matrices. Int. Math. Res. Not. IMRN. (3), 436–479 (2010).Google Scholar
  14. 14.
    Firk, FWK, Miller, SJ: Nuclei, Primes and the Random Matrix Connection. Symmetry. 1, 64–105 (2009). doi:10.3390/sym1010064.MathSciNetCrossRefGoogle Scholar
  15. 15.
    Hayes, B: The spectrum of Riemannium. Am. Sci. 91(4), 296–300 (2003).CrossRefGoogle Scholar
  16. 16.
    Hewitt, E, Ross, K: Abstract harmonic analysis, Vol. I. Structure of topological groups, integration theory, group representations (second edition), fundamental principles of mathematical sciences, Vol. 115. Springer-Verlag, Berlin–New York (1979).Google Scholar
  17. 17.
    Hill, T: The first-digit phenomenon. Am.Sci. 86, 358–363 (1996).CrossRefGoogle Scholar
  18. 18.
    Hill, T: A statistical derivation of the significant-digit law. Stat. Sci. 10, 354–363 (1996).Google Scholar
  19. 19.
    Hurlimann, W: Benford’s law from 1881 to 2006.
  20. 20.
    Katz, N, Sarnak, P: Random matrices, Frobenius eigenvalues and Monodromy, Vol. 45. AMS, Providence (1999).MATHGoogle Scholar
  21. 21.
    Katz, N, Sarnak, P: Zeros of zeta functions and symmetries. Bull. AMS. 36, 1–26 (1999).MATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Keating, JP, Snaith, NC: Random matrix theory and ζ(1/2+i t). Comm. Math. Phys. 214(1), 57–89 (2000).MATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Keating, JP, Snaith, NC: Random matrix theory and L-functions at s=1/2. Comm. Math. Phys. 214(1), 91–110 (2000).MATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    Keating, JP, Snaith, NC: Random matrices and L-functions, random matrix theory. J. Phys. A. 36(12), 2859–2881 (2003).MATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Kontorovich, A, Miller, SJ: Benford’s law, values of L-functions and the 3x+1 problem. Acta Arith. 120, 269–297 (2005).MATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    Knuth, D: The art of computer programming, Volume 2: seminumerical algorithms. 3rd edition. Addison-Wesley, MA (1997).Google Scholar
  27. 27.
    Lagarias, J, Soundararajan, K: Benford’s law for the 3x+1 function. J. London Math. Soc. 74, 289–303 (2006). ser. 2 no. 2.MATHMathSciNetCrossRefGoogle Scholar
  28. 28.
    Mebane, W: Detecting attempted election theft: vote counts, voting machines and Benford’s law. Prepared for delivery at the 2006 Annual meeting of the Midwest Political Science Association, April 20-23. Palmer House, Chicago.Google Scholar
  29. 29.
    Mezzadri, F: How to generate random matrices from the classical compact groups. Notices of the AMS. 54, 592–604 (2007).MATHMathSciNetGoogle Scholar
  30. 30.
    Miller, SJ, (Ed): Benford’s law: theory and applications. Princeton University Press, Princeton, NJ (2015).Google Scholar
  31. 31.
    Miller, SJ, Novikoff, T, Sabelli, A: The distribution of the second largest eigenvalue in families of random regular graphs. Exp. Math. 17(2), 231–244 (2008).MATHCrossRefGoogle Scholar
  32. 32.
    Montgomery, H: The pair correlation of zeros of the zeta function, Analytic number theory. Proc. Sympos. Pure Math. Amer. Math. Soc. Providence. 24, 181–193 (1973).CrossRefGoogle Scholar
  33. 33.
    Newcomb, S: Note on the frequency of use of the different digits in natural numbers. Amer. J. Math. 4, 39–40 (1881).MATHMathSciNetCrossRefGoogle Scholar
  34. 34.
    Nigrini, M: Using digital frequencies to detect fraud. The White Paper. 8(2), 3–6 (1994).Google Scholar
  35. 35.
    Odlyzko, A: On the distribution of spacings between zeros of the zeta function. Math. Comp. 48(177), 273–308 (1987).MATHMathSciNetCrossRefGoogle Scholar
  36. 36.
    Odlyzko, A: The 1022-nd zero of the Riemann zeta function. In: van Frankenhuysen, M, Lapidus, ML (eds.)Proc. Conference on dynamical, Spectral and Arithmetic zeta-functions, p. 2001. Amer. Math. Soc., Contemporary Math. series, San Antonio, TX.Google Scholar
  37. 37.
    Pinkham, R: On the distribution of first significant digits. The Ann. Math. Stat. 32(4), 1223–1230 (1961).MATHMathSciNetCrossRefGoogle Scholar
  38. 38.
    Raimi, RA: The first digit problem. Amer. Math. Monthly. 83(7), 521–538 (1976).MATHMathSciNetCrossRefGoogle Scholar
  39. 39.
    Tao, T, Vu, V: From the littlewood-offord problem to the circular law: universality of the spectral distribution of random matrices. Bull. Amer. Math. Soc. 46, 377–396 (2009).MATHMathSciNetCrossRefGoogle Scholar
  40. 40.
    Tao, T, Vu, V: Random matrices: universality of local eigenvalue statistics up to the edge. Comm. Math. Phys. 298(2), 549–572 (2010).MATHMathSciNetCrossRefGoogle Scholar
  41. 41.
    Tracy, CA, Widom, H: The Distributions of Random Matrix Theory and their Applications. In: Sidoravičius, V (ed.)New Trends in Mathematical Physics. Selected Contributions on the 15th International Congress on Mathematical Physics, pp. 753–765. Springer-Verlag, Netherlands (2009).Google Scholar
  42. 42.
    Varadarajan, VS: Lie groups, Lie algebras and their representations. Prentice-Hall, New York (1974).MATHGoogle Scholar

Copyright information

© The Author(s) 2015

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of MathematicsFranklin & MarshallLancasterUSA
  2. 2.Department of Mathematics and StatisticsWilliams CollegeWilliamstownUSA

Personalised recommendations