Adhikari, AK (1969). Some Results on Distribution of Most Significant Digit. Sankhya-The Indian Journal of Statistics Series B, 31 (Dec), pp. 413-420. ISSN/ISBN:0581-5738.
|
|
|
|
|
Allaart, PC (1997). An invariant-sum characterization of Benford's law. Journal of Applied Probability 34(1), pp. 288-291.
|
|
|
|
|
Barlow, JL and Bareiss, EH (1985). On Roundoff Error Distributions in Floating Point and Logarithmic Arithmetic. Computing 34(4), pp. 325-347. ISSN/ISBN:0010-485X. DOI:10.1007/BF02251833.
|
|
|
|
|
Benford, F (1938). The law of anomalous numbers. Proceedings of the American Philosophical Society, Vol. 78, No. 4 (Mar. 31, 1938), pp. 551-572.
|
|
|
|
|
Berger, A (2001). Chaos and Chance. De Gruyter, Berlin and New York.
|
|
|
|
|
Berger, A (2005). Benford’s Law in power-like dynamical systems. Stochastics and Dynamics 5, pp. 587-607. ISSN/ISBN:0219-4937. DOI:10.1142/S0219493705001602.
|
|
|
|
|
Berger, A (2005). Multi-dimensional dynamical systems and Benford's law. Discrete and Continuous Dynamical Systems 13(1), pp. 219-237. ISSN/ISBN:1078-0947. DOI:10.3934/dcds.2005.13.219.
|
|
|
|
|
Berger, A, Bunimovich, LA and Hill, TP (2005). One-dimensional dynamical systems and Benford's law. Transactions of the American Mathematical Society 357(1), pp. 197-219. ISSN/ISBN:0002-9947. DOI:10.1090/S0002-9947-04-03455-5.
|
|
|
|
|
Berger, A and Hill, TP (2007). Newton’s method obeys Benford’s law. American Mathematical Monthly 114 (7), pp. 588-601. ISSN/ISBN:0002-9890.
|
|
|
|
|
Berger, A and Hill, TP (2011). Benford's Law Strikes Back: No Simple Explanation in Sight for Mathematical Gem. The Mathematical Intelligencer 33(1), pp. 85-91. DOI:10.1007/ s00283-010-9182-3.
|
|
|
|
|
Berger, A, Hill, TP, Kaynar, B and Ridder, A (2011). Finite-state Markov Chains Obey Benford's Law. SIAM Journal of Matrix Analysis and Applications 32(3), pp. 665-684. DOI:10.1137/100789890.
|
|
|
|
|
Berger, A and Siegmund, S (2007). On the distribution of mantissae in nonautonomous difference equations. Journal of Difference Equations and Applications 13(8-9), pp. 829-845. ISSN/ISBN:1023-6198. DOI:10.1080/10236190701388039.
|
|
|
|
|
Diaconis, P (2002). G.H. Hardy and Probability ???. Bulletin of the London Mathematical Society 34(4), pp. 385-402. DOI:10.1112/S002460930200111X.
|
|
|
|
|
Drmota, M and Tichy, RF (1997). Sequences, Discrepancies and Applications. Lecture Notes in Mathematics 1651.
|
|
|
|
|
Engel, HA and Leuenberger, C (2003). Benford's law for exponential random variables. Statistics & Probability Letters 63, pp. 361-365. ISSN/ISBN:0167-7152.
|
|
|
|
|
Feldstein, A and Turner, PR (1996). Overflow and underflow in multiplication and division. Applied Numerical Mathematics 21(3), pp. 221-239. ISSN/ISBN:0168-9274. DOI:10.1016/0168-9274(96)00010-4.
|
|
|
|
|
Feller, W (1971). An Introduction to Probability Theory and Its Applications. 2nd ed., J. Wiley (see p 63, vol 2).
|
|
|
|
|
Fewster, RM (2009). A Simple Explanation of Benford's Law. American Statistician 63(1), pp. 26-32. DOI:10.1198/tast.2009.0005.
|
|
|
|
|
Flehinger, BJ (1966). On the Probability that a Random Integer has Initial Digit A. American Mathematical Monthly 73(10), pp. 1056-1061. ISSN/ISBN:0002-9890. DOI:10.2307/2314636.
|
|
|
|
|
Giuliano-Antonini, R and Grekos, G (2005). Regular sets and conditional density: an extension of Benford's law. Colloquium Mathematicum, 103(2), pp. 173–192. DOI:10.4064/cm103-2-3.
|
|
|
|
|
Hamming, R (1970). On the distribution of numbers. Bell Syst. Tech. J. 49(8), pp. 1609-1625. ISSN/ISBN:0005-8580. DOI:10.1002/j.1538-7305.1970.tb04281.x.
|
|
|
|
|
Hill, TP (1995). A Statistical Derivation of the Significant-Digit Law. Statistical Science 10(4), pp. 354-363. ISSN/ISBN:0883-4237.
|
|
|
|
|
Hill, TP (1995). Base-Invariance Implies Benford's Law. Proceedings of the American Mathematical Society 123(3), pp. 887-895. ISSN/ISBN:0002-9939. DOI:10.2307/2160815.
|
|
|
|
|
Knuth, DE (1997). The Art of Computer Programming. pp. 253-264, vol. 2, 3rd ed, Addison-Wesley, Reading, MA.
|
|
|
|
|
Kontorovich, AV and Miller, SJ (2005). Benford's Law, Values of L-functions and the 3x+ 1 Problem. Acta Arithmetica 120(3), pp. 269-297. ISSN/ISBN:0065-1036. DOI:10.4064/aa120-3-4.
|
|
|
|
|
Kuipers, L and Niederreiter, H (1974). Uniform Distribution of Sequences. J. Wiley; newer edition - 2006 from Dover. ISSN/ISBN:0486450198.
|
|
|
|
|
Lagarias, JC and Soundararajan, K (2006). Benford's law for the 3x+1 function. Journal of the London Mathematical Society 74, pp. 289-303. ISSN/ISBN:0024-6107. DOI:10.1112/S0024610706023131.
|
|
|
|
|
Leemis, LM, Schmeiser, BW and Evans, DL (2000). Survival Distributions Satisfying Benford's Law. American Statistician 54(4), pp. 236-241. ISSN/ISBN:0003-1305. DOI:10.2307/2685773.
|
|
|
|
|
Miller, SJ and Nigrini, MJ (2008). Order Statistics and Benford's Law. International Journal of Mathematics and Mathematical Sciences, Art. ID 382948. ISSN/ISBN:0161-1712. DOI:10.1155/2008/382948.
|
|
|
|
|
Morrison, KE (2010). The Multiplication Game. Mathematics Magazine 83, pp. 100-110. ISSN/ISBN:0025-570X. DOI:10.4169/002557010X482862.
|
|
|
|
|
Newcomb, S (1881). Note on the frequency of use of the different digits in natural numbers. American Journal of Mathematics 4(1), pp. 39-40. ISSN/ISBN:0002-9327. DOI:10.2307/2369148.
|
|
|
|
|
Nigrini, MJ (1992). The Detection of Income Tax Evasion Through an Analysis of Digital Frequencies. PhD thesis, University of Cincinnati, OH, USA.
|
|
|
|
|
Pinkham, RS (1961). On the Distribution of First Significant Digits. Annals of Mathematical Statistics 32(4), pp. 1223-1230. ISSN/ISBN:0003-4851.
|
|
|
|
|
Raimi, RA (1976). The First Digit Problem. American Mathematical Monthly 83(7), pp. 521-538. ISSN/ISBN:0002-9890. DOI:10.2307/2319349.
|
|
|
|
|
Raimi, RA (1985). The First Digit Phenomenon Again. Proceedings of the American Philosophical Society 129(2), pp. 211-219. ISSN/ISBN:0003-049X.
|
|
|
|
|
Schatte, P (1981). On random variables with logarithmic mantissa distribution relative to several bases. Elektronische Informationsverarbeitung und Kybernetik 17(4/6), 293-295. ISSN/ISBN:0013-5712.
|
|
|
|
|
Schatte, P (1988). On a law of the iterated logarithm for sums mod 1 with application to Benford's law. Probability Theory and Related Fields 77(2), 167-178. ISSN/ISBN:0178-8051. DOI:10.1007/BF00334035.
|
|
|
|
|
Schürger, K (2008). Extensions of Black-Scholes processes and Benford's law. Stochastic Processes and their Applications 118(7), 1219-1243. ISSN/ISBN:0304-4149. DOI:10.1016/j.spa.2007.07.017.
|
|
|
|
|
Smith, SW (1997). Explaining Benford's Law. Chapter 34 in: The Scientist and Engineer's Guide to Digital Signal Processing. California Technical Publishing: San Diego, CA. Republished in softcover by Newnes, 2002. ISSN/ISBN:0-9660176-3-3.
|
|
|
|
|
Whitney, RE (1972). Initial digits for the sequence of primes. American Mathematical Monthly 79(2), pp. 150-152. ISSN/ISBN:0002-9890.
|
|
|
|
|