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A basic theory of Benford’s law. (English) Zbl 1245.60016
Summary: Drawing from a large, diverse body of work, this survey presents a comprehensive and unified introduction to the mathematics underlying the prevalent logarithmic distribution of significant digits and significands, often referred to as Benford’s law (BL) or, in a special case, as the first digit law. The invariance properties that characterize BL are developed in detail. Special attention is given to the emergence of BL in a wide variety of deterministic and random processes. Though mainly expository in nature, the article also provides strengthened versions of and simplified proofs for many key results in the literature. Numerous intriguing problems for future research arise naturally.
Reviewer: Reviewer (Berlin)

##### MSC:
 60E05 Probability distributions: general theory 60-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to probability theory 11K06 General theory of distribution modulo $$1$$ 39A60 Applications of difference equations 37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010) 60F15 Strong limit theorems
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##### References:
 [1] Adhikari, A.K. and Sarkar, B.P. (1968), Distributions of most significant digit in certain functions whose arguments are random variables, Sankhya-The Indian Journal of Statistics Series B 30 , 47-58. [2] Allaart, P.C. (1997), An invariant-sum characterization of Benford’s law, J. Appl. Probab. 34 , 288-291. · Zbl 0874.60016 · doi:10.2307/3215195 [3] Barlow, J.L. and Bareiss, E.H. (1985), On Roundoff Error Distributions in Floating Point and Logarithmic Arithmetic, Computing 34 , 325-347. · Zbl 0556.65036 · doi:10.1007/BF02251833 [4] Benford, F. (1938), The law of anomalous numbers, Proc. Amer. Philosophical Soc. 78 , 551-572. · Zbl 0018.26502 [5] Berger, A. (2001), Chaos and Chance , deGruyter, Berlin. [6] Berger, A. (2005), Multi-dimensional dynamical systems and Benford’s Law, Discrete Contin. Dyn. Syst. 13 , 219-237. · Zbl 1075.37003 · doi:10.3934/dcds.2005.13.219 [7] Berger, A. (2005), Benford’s Law in power-like dynamical systems, Stoch. Dyn. 5 , 587-607. · Zbl 1122.37008 · doi:10.1142/S0219493705001602 [8] Berger, A. (2010), Some dynamical properties of Benford sequences, to appear in J. Difference Equ. Appl. · Zbl 1218.11072 · doi:10.1080/10236198.2010.549012 [9] Berger, A. (2010), Large spread does not imply Benford’s law, [10] Berger, A., Bunimovich, L. and Hill, T.P. (2005), Onde-dimensional dynamical systems and Benford’s Law, Trans. Amer. Math. Soc. 357 , 197-219. · Zbl 1123.37006 · doi:10.1090/S0002-9947-04-03455-5 [11] Berger, A. and Hill, T.P. (2007), Newton’s method obeys Benford’s law, Amer. Math. Monthly 114 , 588-601. · Zbl 1136.65048 [12] Berger, A. and Hill, T.P. (2009), Benford Online Bibliography ; accessed May 15, 2011 at . · www.benfordonline.net [13] Berger, A. and Hill, T.P. (2011), Benford’s Law strikes back: No simple explanation in sight for mathematical gem, Math. Intelligencer 33 , 85-91. · Zbl 1221.60010 · doi:10.1007/s00283-010-9182-3 [14] Berger, A., Hill, T.P., Kaynar, B. and Ridder, A. (2011), Finite-state Markov Chains Obey Benford’s Law, to appear in SIAM J. Matrix Analysis . · Zbl 1241.11078 · doi:10.1137/100789890 [15] Berger, A. and Siegmund, S. (2007), On the distribution of mantissae in nonautonomous difference equations, J. Difference Equ. Appl. 13 , 829-845. · Zbl 1130.37005 · doi:10.1080/10236190701388039 [16] Chow, Y.S. and Teicher, H. (1997), Probability Theory. Independence, Interchangeability, Martingales (3rd ed.), Springer. · Zbl 0891.60002 [17] Diaconis, P. (1977), The Distribution of Leading Digits and Uniform Distribution Mod 1, Ann. Probab. 5 , 72-81. · Zbl 0364.10025 · doi:10.1214/aop/1176995891 [18] Drmota, M. and Tichy, R.F. (1997), Sequences, Discrepancies and Applications , Springer. · Zbl 0877.11043 [19] Dubins, L. and Freedman, D. (1967), Random distribution functions, Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66) Vol. II: Contributions to Probability Theory, Part 1, 183-214, Univ. California Press, Berkeley, Calif. [20] Einsiedler, M. (2009), What is measure rigidity? Notices Amer. Math. Soc. 56 600-601. · Zbl 1160.37311 · www.ams.org [21] Engel, H. and Leuenberger, C. (2003), Benford’s law for exponential random variables, Statist. Probab. Lett. 63 , 361-365. · Zbl 1116.60315 · doi:10.1016/S0167-7152(03)00101-9 [22] Feldstein, A. and Turner, P. (1986), Overflow, Underflow, and Severe Loss of Significance in Floating-Point Addition and Subtraction, IMA J. Numer. Anal. 6 , 241-251. · Zbl 0593.65029 · doi:10.1093/imanum/6.2.241 [23] Feller, W. (1966), An Introduction to Probability Theory and Its Applications vol 2, 2nd ed., J. Wiley, New York. · Zbl 0138.10207 [24] Fewster, R. (2009), A simple Explanation of Benford’s Law, Amer. Statist. 63 (1), 20-25. · Zbl 05680821 · doi:10.1198/tast.2009.0005 [25] Flehinger, B.J. (1966), On the Probability that a Random Integer has Initial Digit A, Amer. Math. Monthly 73 , 1056-1061. · Zbl 0147.17502 · doi:10.2307/2314636 [26] Giuliano Antonioni, R. and Grekos, G. (2005), Regular sets and conditional density: an extension of Benford’s law, Colloq. Math. 103 , 173-192. · Zbl 1092.11009 · doi:10.4064/cm103-2-3 [27] Hamming, R. (1970), On the distribution of numbers, Bell Syst. Tech. J. 49 (8), 1609-1625. · Zbl 0211.46701 · doi:10.1002/j.1538-7305.1970.tb04281.x [28] Hill, T.P. (1995), Base-Invariance Implies Benford’s Law, Proc. Amer. Math. Soc. 123 (3), 887-895. · Zbl 0813.60002 [29] Hill, T.P. (1995), A Statistical Derivation of the Significant-Digit Law, Statis. Sci. 10 (4), 354-363. · Zbl 0955.60509 [30] Kallenberg, O. (1997), Foundations of modern probability , Springer. · Zbl 0892.60001 [31] Katok, A. and Hasselblatt, B. (1995), Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge. · Zbl 0878.58020 [32] Knuth, D. (1997), The Art of Computer Programming , pp 253-264, vol. 2, 3rd ed, Addison-Wesley, Reading, MA. [33] Kontorovich, A.V. and Miller, S.J. (2005), Benford’s Law, Values of L-functions and the 3x+1 Problem, Acta Arithm. 120 (3), 269-297. · Zbl 1139.11033 · doi:10.4064/aa120-3-4 [34] Kupiers, L. and Niederreiter, H. (1974), Uniform distribution of sequences , John Wiley & Sons, New York. · Zbl 0281.10001 [35] Lacey, M. and Phillip, W. (1990), A note on the almost sure central limit theorem, Statist. Probab. Lett. 63 , 361-365. [36] Lagarias, J.C. and Soundararajan, K. (2006), Benford’s law for the 3 x +1 function, J. London Math. Soc. 74 , 289-303. · Zbl 1117.11018 · doi:10.1112/S0024610706023131 [37] Leemis, L.M., Schmeiser, B.W. and Evans, D.L. (2000), Survival Distributions Satisfying Benford’s Law, Amer. Statist. 54 (4), 236-241. [38] Lyons, R. (1995), Seventy years of Rajchman measures, J. Fourier Anal. Appl. , Kahane Special Issue, 363-377. · Zbl 0886.43001 [39] Miller, S.J. and Nigrini, M.J. (2008), Order Statistics and Benford’s Law, to appear in: Int. J. Math. Math. Sci . · Zbl 05534756 [40] Morrison, K.E. (2010), The Multiplication Game, Math. Mag. 83 , 100-110. · Zbl 1227.91015 · doi:10.4169/002557010X482862 [41] Newcomb, S. (1881), Note on the frequency of use of the different digits in natural numbers, Amer. J. Math. 9 , 201-205. · JFM 13.0161.01 [42] Nigrini, M.J. (1992), The Detection of Income Tax Evasion Through an Analysis of Digital Frequencies , PhD thesis, University of Cincinnati, OH, USA. [43] Palmer, K. (2000), Shadowing in dynamical systems , Kluwer. · Zbl 0997.37001 [44] Pinkham, R. (1961), On the Distribution of First Significant Digits, Ann. Math. Statist. 32 (4), 1223-1230. · Zbl 0102.14205 · doi:10.1214/aoms/1177704862 [45] Raimi, R. (1976), The First Digit Problem, Amer. Math. Monthly 83 (7), 521-538. · Zbl 0349.60014 · doi:10.2307/2319349 [46] Raimi, R. (1985), The First Digit Phenomenon Again, Proc. Amer. Philosophical Soc. 129 , 211-219. [47] Robbins, H. (1953), On the equidistribution of sums of independent random variables, Proc. Amer. Math. Soc. 4 , 786-799. · Zbl 0053.26704 · doi:10.2307/2032412 [48] Schatte, P. (1988), On random variables with logarithmic mantissa distribution relative to several bases, Elektron. Informationsverarbeit. Kybernetik 17 , 293-295. · Zbl 0476.60021 [49] Schatte, P. (1988), On a law of the iterated logarithm for sums mod 1 with application to Benford’s law, Probab. Theory Related Fields 77 , 167-178. · Zbl 0619.60032 · doi:10.1007/BF00334035 [50] Schürger, K. (2008), Extensions of Black-Scholes processes and Benford’s law, Stochastic Process. Appl. 118 , 1219-1243. · Zbl 1152.60027 · doi:10.1016/j.spa.2007.07.017 [51] Schürger, K. (2011), Lévy Processes and Benford’s Law, [52] Serre, J.P. (1973), A course in arithmetic , Springer. · Zbl 0256.12001 [53] Shiryayev, A.N. (1984), Probability , Springer. [54] Smith, S.W. (1997), Explaining Benford’s Law, Chapter 34 in: The Scientist and Engineer’s Guide to Digital Signal Processing . Republished in softcover by Newnes, 2002 [55] Walter, W. (1998), Ordinary Differential Equations , Springer. · Zbl 0991.34001 [56] Whitney, R.E. (1972), Initial digits for the sequence of primes, Amer. Math. Monthly 79 (2), 150-152. · Zbl 0227.10047 · doi:10.2307/2316536
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