An Invitation to Modern Number Theory

An Invitation to Modern Number Theory

Volumn , Issue , 2020, Pages 1-504

  Miller, Steven J ^a   Takloo Bighash, Ramin ^a  

^a NONE

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EID: 85194309799     PISSN: None     EISSN: None     Source Type: Book    
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References (330)

1. A. Aczel, Fermat’s Last Theorem: Unlocking the Secret of an Ancient Math- ematical Problem, Four Walls Eight Windows, New York, 1996.
2. R. Adler, M. Keane, and M. Smorodinsky, A construction of a normal num- ber for the continued fraction transformation, J. Number Theory 13 (1981), no. 1, 95–105.
3. M. Agrawal, N. Kayal and N. Saxena, PRIMES is in P, Ann. of Math. (2) 160 (2004), no. 2, 781–793.
4. L. Ahlfors, Complex Analysis, 3rd edition, McGraw-Hill, New York, 1979.
5. M. Aigner and G. M. Ziegler, Proofs from THE BOOK, Springer-Verlag, Berlin, 1998.
6. W. R. Alford, A. Granville, and C. Pomerance, There are infinitely many Carmichael numbers, Ann. Math. 139 (1994), 703–722.
7. AMS MathSciNet, http://www.ams.org/msnmain?screen=Review
8. U. Andrews IV and J. Blatz, Distribution of digits in the continued fraction representations of seventh degree algebraic irrationals, Junior Thesis, Prince- ton University, Fall 2002.
9. R. Apéry, Irrationalité de ζ(2) et ζ(3), Astérisque 61 (1979) 11–13.
10. T. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, New York, 1998.
11. S. Arms, A. Lozano-Robledo and S. J. Miller, Constructing One- Parameter Families of Elliptic Curves over Q(T) with Moderate Rank, preprint.
12. M. Artin, Algebra, Prentice-Hall, Englewood Cliffs, NJ, 1991.
13. R. Ayoub, Introduction to the Analytic Theory of Numbers, AMS, Providence, RI, 1963.
14. Z. Bai, Methodologies in spectral analysis of large-dimensional random ma- trices, a review, Statist. Sinica 9 (1999), no. 3, 611–677.
15. A. Baker, Transcendental Number Theory, Cambridge University Press, Cam- bridge, 1990.
16. R. Balasubramanian and C. J. Mozzochi, Siegel zeros and the Goldbach problem, J. Number Theory 16 (1983), no. 3, 311–332.
17. K. Ball and T. Rivoal, Irrationalité d’une infinité valeurs de la fonction zeta aux entiers impairs, Invent. Math. 146 (2001), 193–207.
18. V. V. Batyrev and Yu. Tschinkel, Tamagawa numbers of polarized algebraic varieties, Nombre et répartition de points de hauteur bornée (Paris, 1996), Astérisque (1998), No. 251, 299–340.
19. P. Baxandall and H. Liebeck, Vector Calculus, Clarendon Press, Oxford, 1986.
20. R. Beals, Notes on Fourier series, Lecture Notes, Yale University, 1994.
21. M. Beceanu, Period of the continued fraction of √n, Junior Thesis, Prince-ton University, 2003.
22. F. Benford, The law of anomalous numbers, Proceedings of the American Philosophical Society 78 (1938) 551–572.
23. A. Berger, Leonid A. Bunimovich, and T. Hill, One-dimensional dynamical systems and Benford’s Law, Trans. Amer. Math. Soc. 357 (2005), no. 1, 197– 219.
24. B. Berndt, R. Evans, and K. Williams, Gauss and Jacobi Sums, Canadian Mathematical Society Series of Monographs and Advanced Texts, Vol. 21, Wiley-Interscience Publications, John Wiley & Sons, New York, 1998.
25. M. Bernstein, Games, hats, and codes, lecture at the SUMS 2005 Conference.
26. P. Bickel and K. Doksum, Mathematical Statistics: Basic Ideas and Selected Topics, Holden-Day, San Francisco, 1977.
27. P. Billingsley, Probability and Measure, 3rd edition, Wiley, New York, 1995.
28. P. Bleher, The energy level spacing for two harmonic oscillators with golden mean ratio of frequencies, J. Stat. Phys. 61 (1990) 869–876.
29. P. Bleher, The energy level spacing for two harmonic oscillators with generic ratio of frequencies, J. Stat. Phys. 63 (1991), 261–283.
30. J. Bober, On the randomness of modular inverse mappings, Undergraduate Mathematics Laboratory report, Courant Institute, NYU, 2002.
31. B. Bollobás, Random Graphs, Cambridge Studies in Advanced Mathemat- ics, Cambridge University Press, Cambridge, 2001.
32. E. Bombieri and J. Lagarias, Complements to Li’s criterion for the Rie- mann ypothesis, J. Number Theory 77 (1999), no. 2, 274–287.
33. E. Bombieri and A. van der Poorten, Continued fractions of algebraic num- bers. Pages 137–152 in Computational Algebra and Number Theory (Sydney, 1992), Mathematical Applications, Vol. 325, Kluwer Academic, Dordrecht, 1995.
34. D. Boneh, Twenty years of attacks on the RSA cryptosystem, Notices of the American Mathematical Society, 46 (1999), no. 2, 203–213.
35. Z. Borevich and I. Shafarevich, Number Theory, Academic Press, New York, 1968.
36. J. Borwein and P. Borwein, Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity, John Wiley and Sons, New York, 1987.
37. A. Boutet de Monvel and A. Khorunzhy, Some elementary results around the Wigner semicircle law, lecture notes.
38. W. Boyce and R. DiPrima, Elementary differential equations and boundary value problems, 7th edition, John Wiley & Sons, New York, 2000.
39. R. Brent, The distribution of small gaps between successive primes, Math. Comp. 28 (1974), 315–324.
40. R. Brent, Irregularities in the distribution of primes and twin primes, Col- lection of articles dedicated to Derrick Henry Lehmer on the occasion of his seventieth birthday, Math. Comp. 29 (1975), 43–56.
41. R. Brent, A. van der Poorten, and H. Te Riele, A comparative study of algo- rithms for computing continued fractions of algebraic numbers. Pages 35–47 in Algorithmic number theory (Talence, 1996), Lecture Notes in Computer Science, Vol. 1122, Springer, Berlin, 1996.
42. R. de la Bretèche, Sur le nombre de points de hauteur bornée d’une certaine surface cubique singulière. Pages 51–77 in Nombre et répartition de points de hauteur bornée (Paris, 1996), Astérisque, (1998) no. 251, 51–77.
43. R. de la Bretèche, T. D. Browning, and U. Derenthal, On Manin’s conjec- ture for a certain singular cubic surface, preprint.
44. B. Brindza, A. Pintér, A. van der Poorten, and M. Waldschmidt, On the distribution of solutions of Thue’s equation. Pages 35–46 in Number theory in progress (Zakopane-Koscielisko, 1997), Vol. 1, de Gruyter, Berlin, 1999.
45. T. Brody, J. Flores, J. French, P. Mello, A. Pandey, and S. Wong, Random-matrix physics: spectrum and strength fluctuations, Rev. Mod. Phys. 53 (1981), no. 3, 385–479.
46. J. Brown and R. Duncan, Modulo one uniform distribution of the sequence of logarithms of certain recursive sequences, Fibonacci Quarterly 8 (1970) 482–486.
47. T. Browning, The density of rational points on a certain singular cubic sur- face, preprint.
48. W. Bryc, A. Dembo, T. Jiang, Spectral measure of large random Hankel, Markov and Toeplitz matrices, preprint.
49. A. Bryuno, Continued frations of some algebraic numbers, U.S.S.R. Com- put. Math. & Math. Phys. 4 (1972), 1–15.
50. E. Burger, Exploring the Number Jungle: A Journey into Diophantine Analy- sis, AMS, Providence, RI, 2000.
51. E. Burger and A. van der Poorten, On periods of elements from real quadratic number fields. Pages 35–43 in Constructive, Experimental, and Nonlinear Analysis (Limoges, 1999), CMS Conf. Proc., 27, AMS, Providence, RI, 2000.
52. G. Casella and R. Berger, Statistical Inference, 2nd edition, Duxbury Ad- vanced Series, Pacific Grove, CA, 2002.
53. G. Casati, I. Guarneri, and F. M. Izrailev, Statistical properties of the quasi- energy spectrum of a simple integrable system, Phys. Lett. A 124 (1987), 263–266.
54. L. Carleson, On the convergence and growth of partial sums of Fourier se- ries, Acta Math. 116 (1966), 135–157.
55. J. W. S. Cassels, An Introduction to Diophantine Approximation, Cambridge University Press, London 1957.
56. D. Champernowne, The construction of decimals normal in the scale of ten, J. London Math. Soc. 8 (1933), 254–260.
57. K. Chang, An experimental approach to understanding Ramanujan graphs, Junior Thesis, Princeton University, Spring 2001.
58. J. R. Chen and T. Z. Wang, On the Goldbach problem, Acta Math. Sinica 32 (1989), 702–718.
59. J. Christiansen, An introduction to the moment problem, lecture notes.
60. J. Cisneros, Waring’s problem, Junior Thesis, Princeton University, Spring 2001.
61. J. Coates and A. Wiles, On the conjecture of Birch and Swinnterton-Dyer, Invent. Math. 39 (1977), 43–67.
62. S. Chatterjee and A. Bose, A new method for bounding rates of convergence of empirical spectral distributions, J. Theoret. Probab. 17 (2004), no. 4, 1003– 1019.
63. H. Cohen and H. W. Lenstra, Jr., Heuristics on class groups of number fields. Pages 33–62 in Number Theory, Lecture Notes in Mathematics, Vol. 1068, Springer-Verlag, Berlin, 33–62.
64. H. Cohen and H. W. Lenstra, Jr., Heuristics on class groups, in Number Theory, Lecture Notes in Mathematics, Vol. 1052, Springer-Verlag, Berlin, 26–36.
65. P. Cohen, The independence of the continuum hypothesis, Proc. Nat. Acad. Sci. U.S.A, 50 (1963), 1143–1148; 51 (1964), 105–110.
66. J. Cohn, The length of the period of simple continued fractions, Pacific Journal of Mathematics, 71 (1977), no. 1, 21–32.
67. J. B. Conrey, L-Functions and random matrices. Pages 331–352 in Math- ematics unlimited — 2001 and Beyond, Springer-Verlag, Berlin, 2001.
68. J. B. Conrey, The Riemann hypothesis, Notices of the AMS, 50 (2003), no. 3, 341–353.
69. B. Conrey, D. Farmer, P. Keating, M. Rubinstein and N. Snaith, Integral moments of L-functions, preprint.
70. J. H. Conway, The weird and wonderful chemistry of audioactive decay. Pages 173–178 in Open Problems in Communications and Computation, ed. T. M. Cover and B. Gopinath, Springer-Verlag, New York, 1987.
71. J. H. Conway and R. Guy, The Book of Numbers, Springer-Verlag, Berlin, 1996.
72. J. H. Conway and N. J. A. Sloane, Lexicographic Codes: Error-Correcting Codes from Game Theory, IEEE Trans. Inform. Theory, 32 (1986), no. 3, 219–235.
73. R. M. Corless, Continued fractions and chaos. Amer. Math. Monthly 99 (1992), no. 3, 203–215.
74. Cornell University, arXiv, http://arxiv.org
75. Cornell University, Project Euclid, http://projecteuclid.org/
76. I. P. Cornfeld, S. V. Fomin, and I. G. Sinai, Ergodic Theory, Grundlehren Der Mathematischen Wissenschaften, Springer-Verlag, Berlin, 1982.
77. H. Davenport, The Higher Arithmetic: An Introduction to the Theory of Numbers, 7th edition, Cambridge University Press, Cambridge, 1999.
78. H. Davenport, Multiplicative Number Theory, 2nd edition, revised by H. Montgomery, Graduate Texts in Mathematics, Vol. 74, Springer-Verlag, New York, 1980.
79. H. Davenport, On the distribution of quadratic residues (mod p), London Math. Soc. 6 (1931), 49–54.
80. H. Davenport, On character sums in finite fields, Acta Math. 71 (1939), 99– 121.
81. H. A. David and H. N. Nagaraja, Order Statistics, 3rd edition, Wiley Inter- science, Hoboken, NJ, 2003.
82. G. Davidoff, P. Sarnak, and A. Valette, Elementary Number Theory, Group Theory, and Ramanujan Graphs, London Mathematical Society, Student Texts, Vol. 55, Cambridge University Press, Cambridge 2003.
83. R. Devaney, An Introduction to Chaotic Dynamical Systems, 2nd edition, Westview Press, Cambridge, MA, 2003.
84. P. Diaconis, Patterns in eigenvalues: The 70th Josiah Williard Gibbs lecture, Bulletin of the American Mathematical Society, 40 (2003), no. 2, 155–178.
85. T. Dimofte, Rational shifts of linearly periodic continued fractions, Junior Thesis, Princeton University, 2003.
86. E. Dueñez and S. J. Miller, The Low Lying Zeros of a GL(4) and a GL(6) family of L-functions, preprint.
87. R. Durrett, Probability: Theory and Examples, 2nd edition, Duxbury Press, 1996.
88. F. Dyson, Statistical theory of the energy levels of complex systems: I, II, III, J. Mathematical Phys., 3 (1962) 140–156, 157–165, 166–175.
89. F. Dyson, The threefold way. Algebraic structure of symmetry groups and ensembles in quantum mechanics, J. Mathematical Phys., 3 (1962) 1199– 1215.
90. G. Edgar, Measure, Topology, and Fractal Geometry, 2nd edition, Springer- Verlag, 1990.
91. H. M. Edwards, Riemann’s Zeta Function, Academic Press, New York, 1974.
92. B. Elias, L. Silberman and R. Takloo-Bighash, On Cayley’s theorem, preprint.
93. W. J. Ellison and F. Ellison, Prime Numbers, John Wiley & Sons, New York, 1985.
94. T. Estermann, On Goldbach’s problem: Proof that almost all even positive integers are sums of two primes, Proc. London Math. Soc. Ser. 2 44 (1938) 307–314.
95. T. Estermann, Introduction to Modern Prime Number Theory, Cambridge University Press, Cambridge, 1961.
96. K. Falconer, Fractal Geometry: Mathematical Foundations and Applica- tions, 2nd edition, John Wiley & Sons, New York, 2003.
97. C. Fefferman, Pointwise convergence of Fourier series, Ann. of Math. Ser. 298 (1973), 551–571.
98. W. Feller, An Introduction to Probability Theory and Its Applications, 2nd edition, Vol. II, John Wiley & Sons, New York, 1971.
99. D. Fishman, Closed form continued fraction expansions of special quadratic irrationals, Junior Thesis, Princeton University, 2003.
100. G. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd edition, Pure and Applied Mathematics, Wiley-Interscience, New York, 1999.
101. P. Forrester, Log-gases and random matrices, book in progress.
102. E. Fouvry, Sur la hauteur des points d’une certaine surface cubique sin- gulière. In Nombre et répartition de points de hauteur bornée (Paris, 1996), Astérisque, (1999) no. 251, 31–49.
103. P. J. Forrester, N. C. Snaith, and J. J. M. Verbaarschot, Developments in Random Matrix Theory. In Random matrix theory, J. Phys. A 36 (2003), no. 12, R1–R10.
104. J. Franklin, Mathematical Methods of Economics: Linear and Nonlinear Pro- gramming, Fixed-Point Theorem, Springer-Verlag, New York, 1980.
105. P. Garrett, Making, Breaking Codes: An Introduction to Cryptography, Prentice-Hall, Englewood Cliffs, NJ, 2000.
106. M. Gaudin, Sur la loi limite de l’espacement des valeurs propres d’une ma- trice aléatoire, Nucl. Phys. 25 (1961) 447–458.
107. A. O. Gelfond, Transcendental and Algebraic Numbers, Dover, New York, 1960.
108. A. Gliga, On continued fractions of the square root of prime numbers, Junior Thesis, Princeton University, 2003.
109. K. Gödel, On Formally Undecidable Propositions of Principia Mathematica and Related Systems, Dover, New York, 1992.
110. D. Goldfeld, The class number of quadratic fields and the conjectures of Birch and Swinnerton-Dyer, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 3, 4 (1976), 624–663.
111. D. Goldfeld, The Elementary proof of the Prime Number Theorem, An His- torical Perspective. Pages 179–192 in Number Theory, New York Seminar 2003, eds. D. And G. Chudnovsky, M. Nathanson, Springer-Verlag, New York, 2004.
112. L. Goldmakher, On the limiting distribution of eigenvalues of large ran- dom regular graphs with weighted edges, American Institute of Mathematics Summer REU, 2003.
113. D. A. Goldston and R. C. Vaughan, On the Montgomery-Hooley asymptotic formula. Pages 117–142 in Sieve Methods, Exponential Sums and their Ap- plications in Number Theory, ed. G. R. H. Greaves, G. Harman, and M. N. Huxley, Cambridge University Press, Cambridge, 1996.
114. M. Golubitsky and V. Guillemin, Stable Mappings and Their Singularities, Graduate Texts in Mathematics, Vol. 14, Springer-Verlag, New York, 1973.
115. R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, Addison-Wesley, Reading, MA, 1988.
116. A. Granville and P. Kurlberg, Poisson statistics via the Chinese remainder theorem, preprint.
117. A. Granville and T. Tucker, It’s as easy as abc, Notices of the AMS, 49 (2002), no. 10, 224–1231.
118. B. Gross and D. Zagier, Heegner points and derivatives of L-series, Invent. Math. 84 (1986), no. 2, 225–320.
119. R. Guy, Unsolved Problems in Number Theory (Problem Books in Mathe- matics), 2nd edition, Springer-Verlag, New York, 1994.
120. C. Hammond and S. J. Miller, Eigenvalue spacing distribution for the en- semble of real symmetric Toeplitz matrices, Journal of Theoretical Probability 18 (2005), no. 3, 537–566.
121. G. H. Hardy and J. E. Littlewood, A new solution of Waring’s problem, Q. J. Math. 48 (1919), 272–293.
122. G. H. Hardy and J. E. Littlewood, Some problems of “Partitio Numerorum.” A new solution of Waring’s problem, Göttingen Nach. (1920), 33–54.
123. G. H. Hardy and J. E. Littlewood, Some problems of “Partitio Numerorum.” On the expression of a number as a sum of primes, Acta Math. 44 (1923), 1–70.
124. G. H. Hardy and J. E. Littlewood, Some problems of “Partitio Numerorum.” Further researches in Waring’s problem, Math. Z., 23 (1925) 1–37.
125. G. H. Hardy and E. Wright, An Introduction to the Theory of Numbers, 5th edition, Oxford Science Publications, Clarendon Press, Oxford, 1995.
126. G. H. Hardy and S. Ramanujan, Asymptotic formulae in combinatorial analysis, Proc. London Math. Soc. 17 (1918), 75–115.
127. 2, Proc. Japan. Acad. Ser. A Math. Sci. 68 (1992), 283–286.
128. B. Hayes, The spectrum of Riemannium, American Scientist, 91 (2003), no. 4, 296–300.
129. R. Heath-Brown, The density of rational points on Cayley’s cubic surface, preprint.
130. H. Heillbronn, On the average length of a class of finite continued fractions. In Number Theory and Analysis (A collection of papers in honor of E. Lan- dau), VEB Deutscher Verlag, Berlin, 1968.
131. D. Hejhal, On the triple correlation of zeros of the zeta function, Internat. Math. Res. Notices (1994), no. 7, 294–302.
132. ter Potenzen (Waringsches Problem), Mat. Annalen 67 (1909), 281– 300.
133. T. Hill, The first-digit phenomenon, American Scientist 86 (1996), 358–363.
134. T. Hill, A statistical derivation of the significant-digit law, Statistical Science 10 (1996), 354–363.
135. M. Hindry and J. Silverman, Diophantine Geometry: An Introduction, Grad- uate Texts in Mathematics, Vol. 201, Springer-Verlag, New York, 2000.
136. K. Hrbacek and T. Jech, Introduction to Set Theory, Pure and Applied Math- ematics, Marcel Dekker, New York, 1984.
137. Hua Loo Keng, Introduction to Number Theory, Springer-Verlag, New York, 1982.
138. C. Hughes and Z. Rudnick, Mock Gaussian behaviour for linear statistics of classical compact groups, J. Phys. A 36 (2003) 2919–2932.
139. J. Hull, Options, Futures, and Other Derivatives, 5th edition, Prentice-Hall, Englewood Cliffs, NJ, 2002.
140. K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Graduate Texts in Mathematics, Vol. 84, Springer-Verlag, New York, 1990.
141. H. Iwaniec, Topics in Classical Automorphic Forms, Graduate Studies in Mathematics, Vol. 17, AMS, Providence, RI, 1997.
142. H. Iwaniec and E. Kowalski, Analytic Number Theory, AMS Colloquium Publications, Vol. 53, AMS, Providence, RI, 2004.
143. H. Iwaniec, W. Luo, and P. Sarnak, Low lying zeros of families of L- functions, Inst. Hautes Études Sci. Publ. Math. 91 (2000), 55–131.
144. H. Iwaniec and P. Sarnak, Dirichlet L-functions at the central point. Pages 941–952 in Number Theory in Progress, (Zakopane-Kos´cielisko, 1997), Vol. 2, de Gruyter, Berlin, 1999.
145. H. Iwaniec and P. Sarnak, The non-vanishing of central values of automor- phic L-functions and Landau-Siegel zeros, Israel J. Math. 120 (2000), 155– 177.
146. D. Jakobson, S. D. Miller, I. Rivin, and Z. Rudnick, Eigenvalue spac- ings for regular graphs. Pages 317–327 in Emerging Applications of Number Theory (Minneapolis, 1996), The IMA Volumes in Mathematics and its Ap- plications, Vol. 109, Springer, New York, 1999.
147. N. Jacobson, Basic Algebra I, 2nd edition, W H Freeman & Co, San Fran- cisco, 1985.
148. R. Jeffrey, Formal Logic: Its Scope and Limits, McGraw-Hill, New York, 1989.
149. S. Kapnick, Continued fraction of cubed roots of primes, Junior Thesis, Princeton University, Fall 2002.
150. N. Katz and P. Sarnak, Random Matrices, Frobenius Eigenvalues and Mon- odromy, AMS Colloquium Publications, Vol. 45, AMS, Providence, RI, 1999.
151. N. Katz and P. Sarnak, Zeros of zeta functions and symmetries, Bull. AMS 36 (1999), 1–26.
152. J. P. Keating and N. C. Snaith, Random matrices and L-functions. In Ran- dom Matrix Theory, J. Phys. A 36 (2003), no. 12, 2859–2881.
153. D. Kelley, Introduction to Probability, Macmillan Publishing Company, London, 1994.
154. A. Y. Khinchin, Continued Fractions, 3rd edition, University of Chicago Press, Chicago, 1964.
155. D. Kleinbock, N. Shah, and A. Starkov, Dynamics of subgroup actions on homogeneous spaces of Lie groups and applications to number theory. Pages 813–930 in Handbook of Dynamical Systems, Vol. 1A, North-Holland, Ams- terdam, 2002.
156. A. Knapp, Elliptic Curves, Princeton University Press, Princeton, NJ, 1992.
157. D. Knuth, The Art of Computer Programming, Volume 2: Seminumerical Algorithms, 3rd edition, Addison-Wesley, MA, 1997.
158. N. Koblitz, Why study equations over finitess fields?, Math. Mag. 55 (1982), no. 3, 144–149.
159. N. Koblitz, Elliptic curve cryptosystems, Math. Comp. 48 (1987), no. 177, 203–209.
160. N. Koblitz, A survey of number theory and cryptography. Pages 217-239 in Number Theory, Trends in Mathematics, Birkhäuser, Basel, 2000.
161. V. Kolyvagin, On the Mordell-Weil group and the Shafarevich-Tate group of modular elliptic curves. Pages 429-436 in Proceedings of the International Congress of Mathematicians (Kyoto, 1990), vols. I and II, Math. Soc. Japan, Tokyo, 1991.
162. A. Kontorovich and S. J. Miller, Benford’s law, values of L-functions and the 3x + 1 problem, preprint.
163. A. Kontorovich and Ya. G. Sinai, Structure theorem for (d, g, h)-maps, Bull. Braz. Math. Soc. (N.S.) 33 (2002), no. 2, 213–224.
164. A. Korselt, Probléme chinois, L’intermédiaire math. 6 (1899), 143–143.
165. T. Koshy, Fibonacci and Lucas Numbers with Applications, Wiley- Interscience, New York, 2001
166. F. Kuan, Digit distribution in the continued fraction of ζ(n), Junior Thesis, Princeton University, Fall 2002.
167. L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, John Wiley & Sons, New York, 1974.
168. P. Kurlberg and Z. Rudnick, The distribution of spacings between quadratic residues, Duke Math. J. 100 (1999), no. 2, 211–242.
169. R. Kuzmin, Ob odnoi zadache Gaussa, Doklady Akad. Nauk, Ser. A (1928), 375–380.
170. J. Lagarias, The 3x + 1 problem and its generalizations. Pages 305-334 in Organic mathematics (Burnaby, BC, 1995), CMS Conf. Proc., vol. 20, AMS, Providence, RI, 1997.
171. J. Lagarias, The 3x+1 problem: An annotated bibliography, preprint.
172. J. Lagarias and K. Soundararajan, Benford’s Law for the 3x + 1 function, preprint.
173. S. Lang, Diophantine Geometry, Interscience Publishers, New York, 1962.
174. S. Lang, Introduction to Diophantine Approximations, Addison-Wesley, Reading, MA, 1966.
175. S. Lang, Undergraduate Algebra, 2nd edition, Springer-Verlag, New York, 1986.
176. S. Lang, Calculus of Several Variables, Springer-Verlag, New York, 1987.
177. S. Lang, Undergraduate Analysis, 2nd edition, Springer-Verlag, New York, 1997.
178. S. Lang, Complex Analysis, Graduate Texts in Mathematics, Vol. 103, Springer-Verlag, New York, 1999.
179. S. Lang and H. Trotter, Continued fractions for some algebraic numbers, J. Reine Angew. Math. 255 (1972), 112–134.
180. R. Larson and B. Farber, Elementary Statistics: Picturing the World, Prentice-Hall, Englewood Cliffs, NJ, 2003.
181. R. Laubenbacher and D. Pengelley, Gauss, Eisenstein, and the "third" proof of the quadratic reciprocity theorem: Ein kleines Schauspiel, Math. Intelli- gencer 16 (1994), no. 2, 67–72.
182. J. Law, Kuzmin’s theorem on algebraic numbers, Junior Thesis, Princeton University, Fall 2002.
183. J. Law, The circle method on the binary goldbach conjecture, Junior The- sis, Princeton University, Spring 2003.
184. R. Lehman, First order spacings of random matrix eigenvalues, Junior The- sis, Princeton University, Spring 2000.
185. H. Lenstra and G. Seroussi, On hats and other covers, 2002, preprint.
186. P. Lévy, Sur les lois de probabilite´ dont dependent les quotients complets et incomplets d’une fraction continue, Bull. Soc. Math. 57 (1929), 178–194.
187. R. Lidl, Mathematical aspects of cryptanalysis. Pages 86–97 in Number Theory and Cryptography (Sydney, 1989), London Mathematical Society Lec- ture Note Series, vol. 154, Cambridge University Press, Cambridge, 1990.
188. 2α, Junior Thesis, Princeton University, Spring 2000.
189. Y. Liu, Statistical behavior of the eigenvalues of random matrices, Junior Thesis, Princeton University, Spring 2000.
190. K. Mahler, Arithmetische Eigenschaften einer Klasse von Dezimalbrüchen, Amsterdam Proc. Konin. Neder. Akad. Wet. 40 (1937), 421–428.
191. E. S. Mahmoodian, Mathematical Olympiads in Iran, Vol. I, Sharif Univer- sity Press, Tehran, Iran, 2002.
192. B. Mandelbrot, The Fractal Geometry of Nature, W. H. Freeman, New York, 1982.
193. 2x modulo one, Int. Math. Res. Not. (2003), no. 39, 2131–2151.
194. R. Martin and W. McMillen, An elliptic curve over Q with rank at least 24, Number Theory Listserver, May 2000.
195. A. Massey, S. J. Miller, and J. Sinsheimer, Eigenvalue spacing distribution for the ensemble of real symmetric palindromic Toeplitz matrices, preprint.
196. B. Mazur, Modular curves and the Eisenstein ideal, IHES Publ. Math. 47 (1977), 33–186.
197. B. Mazur, Rational isogenies of prime degree (with an appendix by D. Goldfeld), Invent. Math. 44 (1978), no. 2, 129–162.
198. B. Mazur, Number Theory as Gadfly, Amer. Math. Monthly, 98 (1991), 593–610.
199. B. McKay, The expected eigenvalue distribution of a large regular graph, Linear Algebra Appl. 40 (1981), 203–216.
200. B. McKay and N. Wormald, The degree sequence of a random graph. I. The models, Random Structures Algorithms 11 (1997), no. 2, 97–117.
201. M. Mehta, On the statistical properties of level spacings in nuclear spec- tra, Nucl. Phys. 18 (1960), 395–419.
202. M. Mehta, Random Matrices, 2nd edition, Academic Press, Boston, 1991.
203. N. Metropolis, The beginning of the Monte Carlo method, Los Alamos Sci-ence, No. 15, Special Issue (1987), 125–130.
204. N. Metropolis and S. Ulam, The Monte Carlo method, J. Amer. Statist. As- soc. 44 (1949), 335–341.
205. M. Michelini, Independence of the digits of continued fractions, Junior Thesis, Princeton University, Fall 2002.
206. M. Michelini, Kuzmin’s extraordinaty zero measure set, Senior Thesis, Princeton University, Spring 2004.
207. N. Miller, Various tendencies of non-Poissonian distributions along subse- quences of certain transcendental numbers, Junior Thesis, Princeton Univer- sity, Fall 2002.
208. N. Miller, Distribution of eigenvalue spacings for band-diagonal matrices, Junior Thesis, Princeton University, Spring 2003.
209. S. D. Miller, A simpler way to show ζ(3) is irrational, preprint.
210. S. J. Miller, 1- and 2-level densities for families of elliptic curves: Evidence for the underlying group symmetries, Compositio Mathematica 140 (2004), no. 4, 952–992.
211. S. J. Miller, Density functions for families of Dirichlet characters, preprint.
212. S. J. Miller, The arithmetic mean and geometric inequality, Class Notes from Math 187/487, The Ohio State University, Fall 2003.
213. S. J. Miller, Differentiating identities, Class Notes from Math 162: Statis- tics, Brown University, Spring 2005.
214. S. J. Miller, The Pythagorean won-loss formula in baseball, preprint.
215. S. J. Miller, Investigations of zeros near the central point of elliptic curve L-functions, to appear in Experimental Mathematics.
216. S. J. Miller, Die battles and order statistics, Class Notes from Math 162: Statistics, Brown University, Spring 2006.
217. S. J. Miller and M. Nigrini, Differences of independent variables and almost Benford behavior, preprint.
218. V. Miller, Use of elliptic curves in cryptography. Pages 417–426 in Advances in cryptology – CRYPTO’85 (Santa Barbara, CA, 1985), Lecture Notes in Computer Science, Vol. 218, Springer-Verlag, Berlin, 1986.
219. J. S. Milne, Elliptic Curves, course notes.
220. S. Minteer, Analysis of Benford’s law applied to the 3x+1 problem, Number Theory Working Group, The Ohio State University, 2004.
221. H. Montgomery, Primes in arithmetic progression, Michigan Math. J. 17 (1970), 33–39.
222. H. Montgomery, The pair correlation of zeros of the zeta function. Pages 181–193 in Analytic Number Theory, Proceedings of Symposia in Pure Math- ematics, vol. 24, AMS, Providence, RI, 1973.
223. D. Moore and G. McCabe, Introduction to the Practice of Statistics, W. H. Freeman and Co., London, 2003.
224. H. Montgomery and K. Soundararajan, Beyond pair correlation. Pages 507– 514 in Paul Erdös and His Mathematics, I (Budapest, 1999), Bolyai Society Mathematical Studies, Vol. 11, János Bolyai Math. Soc., Budapest, 2002.
225. C. J. Mozzochi, An analytic sufficiency condition for Goldbach’s conjec- ture with minimal redundancy, Kyungpook Math. J. 20 (1980), no. 1, 1–9.
226. C. J. Mozzochi, The Fermat Diary, AMS, Providence, RI, 2000.
227. C. J. Mozzochi, The Fermat Proof, Trafford Publishing, Victoria, 2004.
228. R. Murty, Primes in certain arithmetic progressions, Journal of the Madras University, (1988), 161–169.
229. R. Murty, Problems in Analytic Number Theory, Springer-Verlag, New York, 2001.
230. M. R. Murty and V. K. Murty, Non-Vanishing of L-Functions and Applica- tions, Progress in Mathematics, vol. 157, Birkhäuser, Basel, 1997.
231. K. Nagasaka and J. S. Shiue, Benford’s law for linear recurrence sequences, Tsukuba J. Math. 11 (1987), 341–351.
232. W. Narkiewicz, The Development of Prime Number Theory, Springer Mono- graphs in Mathematics, Springer-Verlag, New York, 2000.
233. M. Nathanson, Additive Number Theory: The Classical Bases, Graduate Texts in Mathematics, Springer-Verlag, New York, 1996.
234. J. von Neumann and B. Tuckerman, Continued fraction expansion of 21/3, Math. Tables Aids Comput. 9 (1955), 23–24.
235. T. Nicely, The pentium bug, http://www.trnicely.net/pentbug/pentbug.html
236. T. Nicely, Enumeration to 1014 of the Twin Primes and Brun’s Constant, Virginia J. Sci. 46 (1996), 195–204.
237. M. Nigrini, Digital Analysis and the Reduction of Auditor Litigation Risk. Pages 69–81 in Proceedings of the 1996 Deloitte & Touche/University of Kansas Symposium on Auditing Problems, ed. M. Ettredge, University of Kansas, Lawrence, KS, 1996.
238. M. Nigrini, The Use of Benford’s Law as an Aid in Analytical Procedures, Auditing: A Journal of Practice & Theory, 16 (1997), no. 2, 52–67.
239. I. Niven, H. Zuckerman, and H. Montgomery, An Introduction to the The- ory of Numbers, 5th edition, John Wiley & Sons, New York, 1991.
240. T. Novikoff, Asymptotic behavior of the random 3-regular bipartite graph, Undergraduate Mathematics Laboratory report, Courant Institute, NYU, 2002.
241. A. Odlyzko, On the distribution of spacings between zeros of the zeta func- tion, Math. Comp. 48 (1987), no. 177, 273–308.
242. A. Odlyzko, The 1022-nd zero of the Riemann zeta function. Pages 139–144 in Procreedings of the Conference on Dynamical, Spectral and Arithmetic Zeta Functions, ed. M. van Frankenhuysen and M. L. Lapidus, Contemporary Mathematics Series, AMS, Providence, RI, 2001.
243. T. Okano, A note on the transcendental continued fractions, Tokyo J. Math, 10 (1987), no. 1, 151–156.
244. T. Oliveira e Silva, Verification of the Goldbach conjecture up to 6 1016, NMBRTHRY@listserv.nodak.edu mailing list, Oct. 3, 2003, http://listserv.nodak.edu/scripts/wa.exe?A2=ind0310&L=nmbrthry&P=168 and http://www.ieeta.pt/∼tos/goldbach.html
245. L. Olsen, Extremely non-normal continued fractions, Acta Arith. 108 (2003), no. 2, 191–202.
246. A. van der Poorten, An introduction to continued fractions. Pages 99-138 in Diophantine Analysis (Kensington, 1985), London Mathematical Society Lecture Note Series, Vol. 109, Cambridge University Press, Cambridge, 1986.
247. A. van der Poorten, Notes on continued fractions and recurrence se- quences. Pages 86–97 in Number theory and cryptography (Sydney, 1989), London Mathematical Society Lecture Note Series, Vol. 154, Cambridge Uni- versity Press, Cambridge, 1990.
248. A. van der Poorten, Continued fractions of formal power series. Pages 453–466 in Advances in Number Theory (Kingston, ON, 1991), Oxford Science Publications, Oxford University Press, New York, 1993.
249. A. van der Poorten, Fractions of the period of the continued fraction ex- pansion of quadratic integers, Bull. Austral. Math. Soc. 44 (1991), no. 1, 155–169.
250. A. van der Poorten, Continued fraction expansions of values of the expo- nential function and related fun with continued fractions, Nieuw Arch. Wisk. (4) 14 (1996), no. 2, 221–230.
251. A. van der Poorten, Notes on Fermat’s Last Theorem, Canadian Mathemat- ical Society Series of Monographs and Advanced Texts, Wiley-Interscience, New York, 1996.
252. A. van der Poorten and J. Shallit, Folded continued fractions, J. Number Theory 40 (1992), no. 2, 237–250.
253. A. van der Poorten and J. Shallit, A specialised continued fraction, Canad. J. Math. 45 (1993), no. 5, 1067–1079.
254. C. Porter (editor), Statistical Theories of Spectra: Fluctuations, Academic Press, New York, 1965.
255. R. Pyke, Spacings, J. Roy. Statist. Soc. Ser. B 27 (1965), 395–449.
256. R. Qian and D. Steinhauer, Rational relation conjectures, Junior Thesis, Princeton University, Fall 2003.
257. R. Qian and D. Steinhauer, Eigenvalues of weighted random graphs, Junior Thesis, Princeton University, Spring 2003.
258. R. A. Raimi, The first digit problem, Amer. Math. Monthly 83 (1976), no. 7, 521–538.
259. K. Ramachandra, Lectures on Transcendental Numbers, Ramanujan Institute, Madras, 1969.
260. F. Reif, Fundamentals of Statistical and Thermal Physics, McGraw-Hill, New York, 1965.
261. P. Richter, An investigation of expanders and ramanujan graphs along ran- dom walks of cubic bipartite graphs, Junior Thesis, Princeton University, Spring 2001.
262. R. D. Richtmyer, M. Devaney, and N. Metropolis, Continued fraction of algebraic numbers, Numer. Math. 4 (1962), 68–84.
263. G. F. B. Riemann, Über die Anzahl der Primzahlen unter einer gegebenen Grösse, Monatsber. Königl. Preuss. Akad. Wiss. Berlin, Nov. 1859, 671–680 (see [Ed] for an English translation).
264. R. Rivest, A. Shamir, and L. Adleman, A method for obtaining digital sig- natures and public key cryptosystems, Comm. ACM 21 (1978), 120–126.
265. D. Rockmore, Stalking the Riemann Hypothesis: The Quest to Find the Hidden Law of Prime Numbers, Pantheon, New York, 2005.
266. K. Roth, Rational approximations to algebraic numbers, Mathematika 2 (1955), 1–20.
267. M. Rubinstein, A simple heuristic proof of Hardy and Littlewood’s conjec- ture B, Amer. Math. Monthly 100 (1993), no. 5, 456–460.
268. M. Rubinstein, Low-lying zeros of L-functions and random matrix theory, Duke Math. J. 109 (2001), no. 1, 147–181.
269. M. Rubinstein and P. Sarnak, Chebyshev’s bias, Experiment. Math. 3 (1994), no. 3, 173–197.
270. W. Rudin, Principles of Mathematical Analysis, 3rd edition, International Series in Pure and Applied Mathematics, McGraw-Hill, New York, 1976.
271. Z. Rudnick and P. Sarnak, Zeros of principal L-functions and random matrix theory, Duke J. of Math. 81 (1996), 269–322.
272. Z. Rudnick and P. Sarnak, The pair correlation function of fractional parts of polynomials, Comm. Math. Phys. 194 (1998), no. 1, 61–70.
273. 2α, Invent. Math. 145 (2001), no. 1, 37–57.
274. Z. Rudnick and A. Zaharescu, A metric result on the pair correlation of fractional parts of sequences, Acta Arith. 89 (1999), no. 3, 283–293.
275. Z. Rudnick and A. Zaharescu, The distribution of spacings between frac- tional parts of lacunary sequences, Forum Math. 14 (2002), no. 5, 691–712.
276. P. Sarnak Some applications of modular forms, Cambridge Trusts in Math- emetics, Vol. 99, Cambridge University Press, Cambridge, 1990.
277. D. Schmidt, Prime Spacing and the Hardy-Littlewood Conjecture B, Junior Thesis, Princeton University, Spring 2001.
278. P. Schumer, Mathematical Journeys, Wiley-Interscience, John Wiley & Sons, New York, 2004.
279. J. P. Serre, A Course in Arithmetic, Springer-Verlag, New York, 1996.
280. A. Shidlovskii, Transcendental Numbers, Walter de Gruyter & Co., New York, 1989.
281. J. A. Shohat and J. D. Tamarkin, The Problem of Moments, AMS, Provi- dence, RI, 1943.
282. J. Silverman, The Arithmetic of Elliptic Curves, Graduate Texts in Mathe- matics, Vol. 106, Springer-Verlag, New York, 1986.
283. J. Silverman, A Friendly Introduction to Number Theory, 2nd edition, Prentice-Hall, Englewood Cliffs, NJ, 2001.
284. J. Silverman and J. Tate, Rational Points on Elliptic Curves, Springer-Verlag, New York, 1992.
285. B. Simon, The classical moment problem as a self-adjoint finite difference operator, Adv. Math. 137 (1998), no. 1, 82–203.
286. S. Simon and A. Moustakas, Eigenvalue density of correlated complex ran- dom Wishart matrices, Bell Labs Technical Memo, 2004.
287. S. Skewes, On the difference π(x) Li(x), J. London Math. Soc. 8 (1933), 277–283.
288. N. Sloane, On-Line Encyclopedia of Integer Sequences, http://www.research.att.com/∼njas/sequences/Seis.html
289. N. Snaith, Derivatives of random matrix characteristic polynomials with ap- plications to elliptic curves, preprint.
290. E. Stein and R. Shakarchi, Fourier Analysis: An Introduction, Princeton University Press, Princeton, NJ, 2003.
291. E. Stein and R. Shakarchi, Complex Analysis, Princeton University Press, Princeton, NJ, 2003.
292. E. Stein and R. Shakarchi, Real Analysis: Measure Theory, Integration, and Hilbert Spaces, Princeton University Press, Princeton, NJ, 2005.
293. I. Stewart and D. Tall, Algebraic Number Theory, 2nd edition, Chapman & Hall, London, 1987.
294. Strang, Linear Algebra and Its Applications, 3rd edition, Wellesley- Cambridge Press, Wellesley, MA 1998.
295. K. Stromberg, The Banach-Tarski paradox, Amer. Math. Monthly 86 (1979), no. 3, 151–161.
296. P. Szüsz, On the length of continued fractions representing a rational number with given denominator, Acta Arithmetica 37 (1980), 55–59.
297. C. Taylor, The Gamma function and Kuzmin’s theorem, Junior Thesis, Prince- ton University, Fall 2002.
298. R. Taylor and A. Wiles, Ring-theoretic properties of certain Hecke algebras, Ann. Math. 141 (1995), 553–572.
299. C. Tracy and H. Widom, Correlation functions, cluster functions, and spac- ing distributions for random matrices, J. Statist. Phys., 92 (1998), no. 5–6, 809–835.
300. G. Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, Cambridge University Press, Cambridge, 1995.
301. E. C. Titchmarsh, The Theory of the Riemann Zeta-function, revised by D. R. Heath-Brown, Oxford University Press, Oxford, 1986.
302. R. C. Vaughan, On a variance associated with the distribution of primes in arithmetic progression, Proc. London Math. Soc. (3) 82 (2001), 533–553.
303. R. C. Vaughan and T. D. Wooley, Waring’s problem: A survey. Pages 301– 340 in Number Theory for the Millennium, III (Urbana, IL, 2000), A. K. Peters, Natick, MA, 2002.
304. I. Vinogradov, Representation of an odd number as the sum of three primes, Doklady Akad. Nauk SSSR, 15 (1937), no. 6–7, 291–294.
305. I. Vinogradov, Some theorems concerning the theory of primes, Mat. Sbornik, 2 (1937), no. 44, 179–195.
306. A. Voros, A sharpening of Li’s criterion for the Riemann hypothesis, preprint.
307. W. Voxman and R. Goetschel, Jr., Advanced Calculus, Mercer Dekker, New York, 1981.
308. L. Washington, Elliptic Curves: Number Theory and Cryptography, Chap- man & Hall/CRC, New York, 2003.
309. S. Wedeniwski, ZetaGrid, http://www.zetagrid.net
310. A. Weil, Numbers of Solutions of Equations in Finite Fields, Bull. Amer. Math. Soc. 14 (1949), 497–508.
311. A. Weil, Prehistory of the zeta-function. Pages 1–9 in Number Theory, Trace Formulas and Discrete Groups (Oslo, 1987), Academic Press, Boston, 1989.
312. B. Weir, The local behavior of Germain primes, Undergraduate Mathemat- ics Laboratory report, Courant Institute, NYU, 2002.
313. E. Weisstein, MathWorld — A Wolfram Web Resource, http://mathworld.wolfram.com
314. H. Weyl, The Classical Groups: Their Invariants and Representations, Princeton University Press, Princeton, NJ, 1946.
315. E. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies: With an Introduction to the Problem of Three Bodies, Dover, New York, 1944.
316. E. Whittaker and G. Watson, A Course of Modern Analysis, 4th edition, Cambridge University Press, Cambridge, 1996.
317. E. Wigner, On the statistical distribution of the widths and spacings of nuclear resonance levels, Proc. Cambridge Philo. Soc. 47 (1951), 790–798.
318. E. Wigner, Characteristic vectors of bordered matrices with infinite dimen- sions, Ann. of Math. 2 (1955), no. 62, 548–564.
319. E. Wigner, Statistical Properties of real symmetric matrices. Pages 174– 184 in Canadian Mathematical Congress Proceedings, University of Toronto Press, Toronto, 1957.
320. E. Wigner, Characteristic vectors of bordered matrices with infinite dimen- sions. II, Ann. of Math. Ser. 2 65 (1957), 203–207.
321. E. Wigner, On the distribution of the roots of certain symmetric matrices, Ann. of Math. Ser. 2 67 (1958), 325–327.
322. A. Wiles, Modular elliptic curves and Fermat’s last theorem, Ann. Math. 141 (1995), 443–551.
323. H. Wilf, Algorithms and Complexity, 2nd edition, A. K. Peters, Natick, MA, 2002.
324. E. Wirsing, On the theorem of Gauss-Kuzmin-Lévy and a Frobenius-type theorem for function spaces, Acta Arith. 24 (1974) 507–528.
325. J. Wishart, The generalized product moment distribution in samples from a normal multivariate population, Biometrika 20 A (1928), 32–52.
326. N. C. Wormald, Models of random regular graphs. Pages 239–298 in Sur- veys in combinatorics, 1999 (Canterbury) London Mathematical Society Lec- ture Note Series, vol. 267, Cambridge University Press, Cambridge, 1999.
327. T. Wooley, Large improvements in Waring’s problem, Ann. of Math. (2), 135 (1992), no. 1, 131–164.
328. I. Zakharevich, A generalization of Wigner’s law, preprint.
329. W. Zudilin, One of the numbers ζ(5),ζ(7),ζ(9),ζ(11) is irrational, Uspekhi Mat. Nauk 56 (2001), 149-150.
330. A. Zygmund, Trigonometrical Series, vols. I and II, Cambridge University Press, Cambridge, 1968.