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Original Articles

Leading Digits of Mersenne Numbers

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References

  • [Apostol 76] T. M. Apostol. Introduction to Analytic Number Theory, New York-Heidelberg: Springer-Verlag, 1976, Undergraduate Texts in Mathematics. MR 0434929.
  • [Banks and Shparlinski 09] W. D. Banks and I. E. Shparlinski. “Prime Numbers with Beatty Sequences.” Colloq. Math. 115:2 (2009), 147–157. MR 2491740.
  • [Banks et al. 12] W. D. Banks, J. B. Friedlander, M. Z. Garaev, and I. E. Shparlinski. “Exponential and Character Sums with Mersenne Numbers.” J. Aust. Math. Soc. 92:1 (2012), 1–13. MR 2945673.
  • [Bateman and Diamond 04] P. T. Bateman and H. G. Diamond. Analytic Number Theory, Monographs in Number Theory, vol. 1. Hackensack, NJ: World Scientific Publishing Co. Pvt. Ltd. , 2004, An introductory course. MR 2111739.
  • [Beebe 17] N. H. F. Beebe. A Bibliography of Publications About Benford’s Law, Heap’s Law and Zipf’s Law. 2017. ftp://ftp.math.utah.edu/public_html/public_html/pub/tex/bib/benfords-law.pdf.
  • [Benford 38] F. Benford. “The Law of Anomalous Numbers.” Proc. Am. Philos. Soc. 78:4 (1938), 551–572.
  • [Berger and Hill 11] A. Berger and T. P. Hill. “A Basic Theory of Benford’s Law.” Probab. Surv. 8 (2011), 1–126. MR 2846899.
  • [Berger and Hill 15] A. Berger and T. P. Hill. An Introduction to Benford’s Law. Princeton, NJ: Princeton University Press, 2015. MR 3242822.
  • [Berger et al. 09] A. Berger, T. P. Hill, and E. Rogers, Benford Online Bibliography. 2009. http://www.benfordonline.net.
  • [Cai, Faust et al. 18] Z. Cai, M. Faust, A. J. Hildebrand, J. Li, and Y. Zhang. The Unreasonable Effectiveness of Benford’s Law in Mathematics. Preprint, 2018.
  • [Cai, Hildebrand et al.] Z. Cai, A. J. Hildebrand, and J. Li, A local Benford Law for a Class of Arithmetic Sequences. Int. J. Number Theory, in press. http://arXiv.org/1808.01496.
  • [Chenavier 17] N. Chenavier, B. Massé, and D. Schneider. “Products of Random Variables and the First Digit Phenomenon.” Stoch. Processes Appl. 128:5 (2017), 1615–1634.
  • [Cohen and Katz 84] D. I. A. Cohen and T. M. Katz. “Prime Numbers and the First Digit Phenomenon.” J. Number Theory 18:3 (1984), 261–268. MR 746863.
  • [Diaconis 77] P. Diaconis. “The Distribution of Leading Digits and Uniform Distribution Mod 1.” Ann. Prob. 5:1 (1977), 72–81. MR 0422186.
  • [Eliahou et al. 13] S. Eliahou, B. Massé, and D. Schneider. “On the Mantissa Distribution of Powers of Natural and Prime Numbers.” Acta Math. Hung. 139:1–2 (2013), 49–63. MR 3028653.
  • [Fuchs and Letta 96] A. Fuchs and G. Letta. “Le problème du premier chiffre décimal pour les nombres premiers.” Electron. J. Comb. 3:2 (1996), Research Paper 25, 7 pp. The Foata Festschrift. MR 1392510.
  • [Gallagher 76] P. X. Gallagher. “On the Distribution of Primes in Short Intervals.” Mathematika 23:1 (1976), 4–9. MR 0409385.
  • [Hill 95] T. P. Hill. “The Significant-Digit Phenomenon.” Amer. Math. Monthly 102:4 (1995), 322–327. MR 1328015.
  • [Hürlimann 09a] W. Hürlimann. “Generalizing Benford’s Law Using Power Laws: Application to Integer Sequences.” Int. J. Math. Math. Sci. 2009 (2009)a, 45. MR 2533550.
  • [Hürlimann 09b] W. Hürlimann. Benford’s Law from 1881 to 2006: A Bibliography. 2009b. http://arxiv.org/ftp/math/papers/0607/0607168.pdf.
  • [Iwaniec and Kowalski 04] H. Iwaniec and E. Kowalski. Analytic Number Theory, American Mathematical Society Colloquium Publications, vol. 53, Providence, RI: American Mathematical Society, 2004. MR 2061214.
  • [Kuipers and Niederreiter 74] L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences. New York-London-Sydney: Wiley-Interscience [John Wiley & Sons], 1974. Pure and Applied Mathematics. MR 0419394.
  • [Luque and Lacasa 09] B. Luque and L. Lacasa. “The First-Digit Frequencies of Prime Numbers and Riemann Zeta Zeros.” Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 465:2107 (2009), 2197–2216. MR 2515637.
  • [Maier 85] H. Maier. “Primes in Short Intervals.” Mich. Math. J. 32:2 (1985), 221–225. MR 783576.
  • [Massé and Schneider 11] B. Massé and D. Schneider. “A Survey on Weighted Densities and their Connection with the First Digit Phenomenon.” Rocky Mt. J. Math. 41:5 (2011), 1395–1415. MR 2838069.
  • [Massé and Schneider 14] B. Massé and D. Schneider. “The Mantissa Distribution of the Primorial Numbers.” Acta Arith. 163:1 (2014), 45–58. MR 3194056.
  • [Massé and Schneider 15] B. Massé and D. Schneider. “Fast Growing Sequences of Numbers and the First Digit Phenomenon.” Int. J. Number Theory 11:705 (2015), 705–719. MR 3327839.
  • [Miller 15] S. J. Miller (ed.), Benford’s Law: Theory and Applications. Princeton, NJ: Princeton University Press, 2015. MR 3408774.
  • [Miller and Nigrini 08] S. J. Miller and M. J. Nigrini. “The Modulo 1 Central Limit Theorem and Benford’s Law for Products.” Int. J. Algebra 2:1–4 (2008), 119–130. MR 2417189.
  • [Müller 01] N. Müller. The iRRAM: Exact arithmetic in C++: Computability and Complexity in Analysis. New York: Springer, 2001, pp. 222–252.
  • [Newcomb 81] S. Newcomb. “Note on the Frequency of Use of the Different Digits in Natural Numbers.” Amer. J. Math. 4:1–4 (1881), 39–40. MR 1505286
  • [Nigrini 12] M. Nigrini. Benford’s Law: Applications for Forensic Accounting, Auditing, and Fraud Detection, vol. 586. New York: John Wiley & Sons, 2012.
  • [Raimi 76] R. A. Raimi. “The First Digit Problem.” Amer. Math. Monthly 83:7 (1976), 521–538. MR 0410850.
  • [Schatte 83] P. Schatte. “On H∞-summability and the Uniform Distribution of Sequences.” Math. Nachr. 113 (1983), 237–243. MR 725491.
  • [Soundararajan 07] K. Soundararajan. The Distribution of Prime Numbers, Equidistribution in Number Theory, an Introduction: NATO Science Series II: Mathematics, Physics and Chemistry, vol. 237. Dordrecht: Springer, 2007, pp. 59–83. MR 2290494.
  • [Vinogradov 47] I. M. Vinogradov. “The Method of Trigonometrical Sums in the Theory of Numbers.” Trav. Inst. Math. Stekloff 23 (1947), 109. MR 0029417.
  • [Whitney 72] R. E. Whitney. “Initial Digits for the Sequence of Primes.” Amer. Math. Monthly 79 (1972), 150–152. MR 0304337.

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