Benford's law, values of L-functions and the 3x + 1 problem

Acta Arithmetica

Volumn 120, Issue 3, 2005, Pages 269-297

  Kontorovich, Alex V ^a,b   Miller, Steven J ^a,c,d  

^a Columbia University ^*

^b Och Spine at New York Presbyterian Hospitals   (United States)

^c OHIO STATE UNIVERSITY   (United States)

^d BROWN UNIVERSITY   (United States)

Author keywords

3x + 1; Benford's Law; Equidistribution; Irrationality type; L function; Poisson Summation

Indexed keywords

EID: 33645139599     PISSN: 00651036     EISSN: None     Source Type: Journal    
DOI: 10.4064/aa120-3-4     Document Type: Review

References (35)

1. [BPY] P. Baine, J. Pitman and M. Yor, Probability laws related to the Jacobi theta and Riemann zeta functions, and Brownian excursions, Bull. Amer. Math. Soc. 38 (2001), 435-465.
2. [Ba] A. Baker, The theory of linear forms in logarithms, in: Transcendence Theory: Advances and Applications, A. Baker and D. W. Masser (eds.), Academic Press, 1977, 1-27.
3. [Ben] F. Benford, The law of anomalous numbers, Proc. Amer. Philos. Soc. 78 (1938), 551-572.
4. [BBH] A. Berger, L. A. Bunimovich and T. Hill, One-dimensional dynamical systems and Benford's Law, Trans. Amer. Math. Soc. 357 (2005), 197-219.
5. [Bi] P. Billingsley, Prime numbers and Brownian motion, Amer. Math. Monthly 80 (1973), 1099-1115.
6. [BJ] H. Bohr and B. Jessen, On the distribution of the values of the Riemann zeta-function, Amer. J. Math. 58 (1936), 35-44.
7. [Con] J. B. Conrey, The Riemann Hypothesis, Notices Amer. Math. Soc. 50 (2003), 341-353.
8. [Da] H. Davenport, Multiplicative Number Theory, 2nd ed., revised by H. Montgomery, Grad. Texts in Math. 74, Springer, New York, 1980.
9. [Dia] P. Diaconis, The distribution of leading digits and uniform distribution mod 1, Ann. Probab. 5 (1979), 72-81.
10. [Fe] W. Feller, An Introduction to Probability Theory and its Applications, Vol. II, 2nd ed., Wiley, 1971.
11. [Hej] D. Hejhal, On a result of Selberg concerning zeros of linear combinations of L-functions, Int. Math. Res. Not. 11 (2000), 551-577.
12. [Hi1] T. Hill, The first-digit phenomenon, Amer. Sci. 86 (1996), 358-363.
13. [Hi2] _, A statistical derivation of the significant-digit law, Statist. Sci. 10 (1996), 354-363.
14. [HS] M. Hindry and J. Silverman, Diophantine Geometry: An Introduction, Grad. Texts in Math. 201, Springer, New York, 2000.
15. [Iv] A. Ivić, On small values of the Riemann zeta-function on the critical line and gaps between zeros, Liet. Mat. Rink. 42 (2002), 31-45.
16. [IK], H. Iwaniec and E. Kowalski, Analytic Number Theory, Amer. Math. Soc., Providence, RI, 2004.
17. [JW] B. Jessen and A. Winter, Distribution functions and the Riemann zeta function, Trans. Amer. Math. Soc. 38 (1935), 48-88.
18. [KaSh] I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Grad. Texts in Math. 113, Springer, 1991.
19. [KaSa] N. Katz and P. Sarnak, Zeros of zeta functions and symmetries, Bull. Amer. Math. Soc. 36 (1999), 1-26.
20. [KeSn] J. P. Keating and N. C. Snaith, Random matrix theory and ζ(1/2 + it), Comm. Math. Phys. 214 (2000), 57-89.
21. [Knu] D. Knuth, The Art of Computer Programming, Volume 2: Seminumerical Algorithms, 3rd ed., Addison-Wesley, 1997.
22. [KonSi] A. Kontorovich and Ya. G. Sinai, Structure theorem for (d, g, h)-maps, Bull. Braz. Math. Soc. (N.S.) 33 (2002), 213-224.
23. [KN] L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, Wiley, 1974.
24. [Lag1] J. Lagarias, The 3x+1 problem and its generalizations, in: Organic Mathematics (Burnaby, BC, 1995), CMS Conf. Proc. 20, Amer. Math. Soc., Providence, RI, 1997, 305-334.
25. [Lag2] _, The 3x + 1 problem: An annotated bibliography, http://arxiv.org/abs/math/0309224.
26. [LS] J. Lagarias and K. Soundararajan, Benford's law for the 3x+1 function, preprint.
27. [Lau] A. Laurincikas, Limit theorems for the Riemann zeta-function on the critical line II, Liet. Mat. Rink. 27 (1987), 489-500.
28. [Luo] W. Luo, Zeros of Hecke L-functions associated with cusp forms, Acta Arith. 71 (1995), 139-158.
29. [New] S. Newcomb, Note on the frequency of use of the different digits in natural numbers, Amer. J. Math. 4 (1881), 39-40.
30. [Pin] R. Pinkham, On the distribution of first significant digits, Ann. Math. Statist. 32 (1961), 1223-1230.
31. [Ro] K. Roth, Rational approximations to algebraic numbers, Mathematika 2 (1955), 1-20.
32. [Sel1] A. Selberg, Contributions to the theory of the Riemann zeta-function, Arch. Math. Naturvid. 48 (1946), no. 5, 89-155.
33. [Sel2] -, Old and new conjectures and results about a class of Dirichlet series, in: Proc. Amalfi Conference on Analytic Number Theory, E. Bombieri et al. (eds.), Univ. Salerno, Salerno, 1992, 367-385.
34. [Se] J. P. Serre, A Course in Arithmetic, Springer, 1996.
35. [Sin] Ya. G. Sinai, Statistical (3x + 1) problem, Comm. Pure Appl. Math. 56 (2003), 1016-1028.