An Invitation to Modern Number Theory
Steven J. Miller (Steven.J.Miller AT williams.edu) and Ramin Takloo-Bighash
(rtakloo AT math.uic.edu)
This website has been moved, so if any link doesn't work, please email Steven.J.Miller AT williams.edu
In Spring 2009 several parts of this book were used for an advanced course at Williams College (Math 406: Analysis and Number Theory). Click here for the course homepage, which includes readings for the various sections, homework assignments, and additional comments for each lecture.
Other useful links:
Errata for the book (if you find a typo, please email Steven.J.Miller AT williams.edu).
STUDENT REPORTS: PRINCETON NYU OHIO STATE AMERICAN INSTITUTE OF MATHEMATICS
DOWNLOADABLE PAPERS (From the Bibliography - links slowly being
[AgKaSa] PRIMES is in P (M. Agrawal, N. Kayal and N. Saxena), Ann. of Math. (2) 160 (2004), no. 2, 781--793.
[AGP] There are infinitely many Carmichael numbers (W. R. Alford, A. Granville, and C. Pomerance), Ann. Math. 139 (1994), 703--722.
[AB] Distribution of digits in the continued fraction representations of seventh degree algebraic irrationals (U. Andrews IV and J. Blatz), Junior Thesis, Princeton University, Fall 2002.
[ALM] Constructing One-Parameter Families of Elliptic Curves over Q(T) with Moderate Rank (Scott Arms, Alvaro Lozano-Robledo, Steven J. Miller). Journal of Number Theory 123 (2007), no. 2, 388--402.
[BR] Irrationalite d'une infinite valeurs de la fonction zeta aux entiers impairs (K. Ball and T. Rivoal), Invent. Math. 146 (2001), 193--207.
Period of the continued fraction of sqrt(n) (M. Beceanu), Junior Thesis,
Princeton University, 2003.
[BBH] One-dimensional dynamical systems and Benford's Law (A. Berger, Leonid A. Bunimovich, and T. Hill), Trans. Amer. Math. Soc. 357 (2005), no. 1, 197--219.
[Ber] Games, Hats, and Codes (Mira Bernstein).
[Bob] On the randomness of modular inverse mappings (J. Bober), Undergraduate Mathematics Laboratory report, Courant Institute, NYU, 2002.
[BoLa] Complements to Li's criterion for the Riemann Hypothesis (E. Bombieri and J. Lagarias), J. Number Theory 77 (1999), no. 2, 274--287.
[BP] Continued fractions of algebraic numbers (E. Bombieri and A. van der Poorten). Pages 137--152 in Computational Algebra and Number Theory (Sydney, 1992), Mathematical Applications, Vol. 325, Kluwer Academic, Dordrecht, 1995.
[Bon] Twenty years of attacks on the RSA cryptosystem (D. Boneh), Notices of the American Mathematical Society, 46 (1999), no. 2, 203--213.
[BK] Some elementary results around the Wigner semicircle law (A. Boutet de Monvel and A. Khorunzhy), lecture notes.
[Bre1 The distribution of small gaps between successive primes (R. Brent), Math. Comp. 28 (1974), 315--324.
[Bre2] Irregularities in the distribution of primes and twin primes (R. Brent), Collection of articles dedicated to Derrick Henry Lehmer on the occasion of his seventieth birthday, Math. Comp. 29 (1975), 43--56.
[BPR] A comparative study of algorithms for computing continued fractions of algebraic numbers (R. Brent, A. van der Poorten, and H. te Riele). Pages 35--47 in Algorithmic number theory (Talence, 1996), Lecture Notes in Computer Science, Vol. 1122, Springer, Berlin, 1996.
[BFFMPW] Random-matrix physics: spectrum and strength fluctuations (T. Brody, J. Flores, J. French, P. Mello, A. Pandey, and S. Wong), Rev. Mod. Phys. 53 (1981), no. 3, 385--479.
[Bro] The density of rational points on a certain singular cubic surface (T. Browning), preprint.
[BDJ] Spectral measure of large random Hankel, Markov and Toeplitz matrices (W. Bryc, A. Dembo, T. Jiang), Annals of Probability 34 (2006), no. 1, 1--38
[BuP] On periods of elements from real quadratic number fields (E. Burger and A. van der Poorten). Pages 35--43 in Constructive, Experimental, and Nonlinear Analysis (Limoges, 1999), CMS Conf. Proc., 27, AMS, Providence, RI, 2000.
[Chttp://arxiv.org/pdf/math-ph/0608050GI] Statistical properties of the quasi-energy spectrum of a simple integrable system (G Casati, I. Guarneri, and F. M. Izrailev), Phys. Lett. A 124 (1987), 263--266.
[Cha] An experimental approach to understanding Ramanujan graphs (K. Chang), Junior Thesis, Princeton University, Spring 2001.
[ChWa] On the Goldbach problem (J. R. Chen and T. Z. Wang), Acta Math. Sinica 32 (1989), 702--718.
[Chr] An Introduction to the Moment Problem (Jacob Christiansen).
[Ci] Waring's problem (J. Cisneros), Junior Thesis, Princeton University, Spring 2001.
[CB] A new method for bounding rates of convergence of empirical spectral distributions (S. Chatterjee and A. Bose), J. Theoret. Probab. 17 (2004), no. 4, 1003--1019.
[Cof1] Toward verification of the Riemann hypothesis: Application of the Li criterion, to appear in Math. Physics, Analysis and Geometry.
[Coh] The independence of the continuum hypothesis (P. Cohen), Proc. Nat. Acad. Sci. U.S.A, 50 (1963), 1143--1148; 51 (1964), 105--110.
[Cohn] The length of the period of simple continued fractions (J. Cohn), Pacific Journal of Mathematics, 71 (1977), no. 1, 21--32.
[Con1] L-Functions and random matrices (J. B. Conrey). Pages 331--352 in Mathematics unlimited --- 2001 and Beyond, Springer-Verlag, Berlin, 2001.
[Con2] The Riemann hypothesis (J. B. Conrey), Notices of the AMS, 50 (2003), no. 3, 341--353.
[CFKRS] Integral moments of
L-functions (B. Conrey, D. Farmer, P. Keating, M. Rubinstein and N. Snaith),
Proc. London Math. Soc.
(3) 91 (2005), no. 1, 33--104.
Codes: Error-Correcting Codes from Game Theory (J. H. Conway and N. J. A.
Sloane), IEEE Trans. Inform. Theory, 32
(1986), no. 3, 219--235.
[Corl] Continued fractions and chaos (R. M. Corless), Amer. Math. Monthly 99 (1992), no. 3, 203--215.
[Cor1] arXiv (Cornell University).
[Cor2] Project Euclid (Cornell University).
[Dia] Patterns in eigenvalues: the 70th Josiah Williard Gibbs lecture (P. Diaconis), Bulletin of the American Mathematical Society, 40 (2003), no. 2, 155--178.
[Di] Rational shifts of linearly periodic continued fractions (T. Dimofte), Junior Thesis, Princeton University, 2003.
[DM] The Low Lying Zeros of a GL(4) and a GL(6) family of L-functions (Eduardo Duenez and Steven J. Miller), to appear in Compositio Mathematica.
[Dy2] The threefold way. Algebraic structure of symmetry groups and ensembles in quantum mechanics (F. Dyson), J. Mathematical Phys., 3 (1962) 1199--1215.
[EST] On Cayley's Theorem (B. Elias, L. Silberman and R. Takloo-Bighash).
[Fef] Pointwise convergence of Fourier series (C. Fefferman), Ann. of Math. Ser. 2 98 (1973), 551--571.
[Fi] Closed form continued fraction expansions of special quadratic irrationals (D. Fishman), Junior Thesis, Princeton University, 2003.
[For] Log-gases and Random matrices (Peter Forrester): book in progress
[GP1] Ramanujan-Fourier series and the density of Sophie Germain primes (H. G. Gadiyar and R. Padma), 2005, http://arxiv.org/abs/math/0508639.
GP2] Linking the Circle and the Sieve: Ramanujan - Fourier Series (H. G. Gadiyar and R. Padma), 2006, http://arxiv.org/abs/math/0601574.
[Gl] On continued fractions of the square root of prime numbers (A. Gliga), Junior Thesis, Princeton University, 2003.
[Gol2] The Elementary proof of the Prime Number Theorem, An Historical Perspective (D. Goldfeld). Pages 179--192 in Number Theory, New York Seminar 2003, eds. D. and G. Chudnovsky, M. Nathanson, Springer-Verlag, New York, 2004.
[Gold] On the Limiting Distribution of Eigenvalues of Large Random Regular Graphs with Weighted Edges (Leo Goldmakher).
[GK] Poisson statistics via the Chinese remainder theorem (A. Granville and P. Kurlberg), preprint.
[GT] It's as easy as abc (A. Granville and T. Tucker), Notices of the AMS, 49 (2002), no. 10, 224--1231.
of Eigenvalues for the Ensemble of Real Symmetric Toeplitz Matrices (Chris
Hammond, Steven J. Miller), Journal of
Theoretical Probability 18 (2005), no. 3, 537--566.
[HL1] A new solution of Waring's problem (G. H. Hardy and J. E. Littlewood), Q. J. Math.48 (1919), 272--293.
[HL2] Some problems of ``Partitio Numerorum.'' A new solution of Waring's problem (G. H. Hardy and J. E. Littlewood), Gottingen Nach. (1920), 33--54.
[HL3] Some problems of ``Partitio Numerorum.'' III. On the expression of a number as a sum of primes (G. H. Hardy and J. E. Littlewood), Acta Math. 44 (1923), 1--70.
[HL4] Some problems of ``Partitio Numerorum.'' IV. Further researches in Waring's problem (G. H. Hardy and J. E. Littlewood), Math. Z., 23 (1925) 1--37.
[HR] Asymptotic formulae in combinatorial analysis (G. H. Hardy and S. Ramanujan), Proc. London Math. Soc. 17 (1918), 75--115.
[Ha1] Third Base: Three cheers for base 3! (B. Hayes), American Scientist 89 (2001), no. 6, 490--494
[Ha2] The spectrum of Riemannium (B. Hayes), American Scientist 91 (2003), no. 4, 296--300.
[He] The density of rational points on Cayley's cubic surface (R. Heath-Brown), preprint.
[Hej] On the triple correlation of zeros of the zeta function (D. Hejhal), Internat. Math. Res. Notices (1994), no. 7, 294--302.
[Hi1] The first-digit phenomenon (T. Hill), American Scientist 86 (1996), 358--363.
[Hi2] A statistical derivation of the significant-digit law (T. Hill), Statistical Science 10 (1996), 354--363.
[HuRu] Mock Gaussian behaviour
for linear statistics of classical compact groups (C. Hughes and Z.
Rudnick), J. Phys. A 36 (2003)
[ILS] Low lying zeros of families of L-functions (H. Iwaniec, W. Luo, and P. Sarnak), Inst. Hautes Etudes Sci. Publ. Math. 91 (2000), 55--131.
[JMRR] Eigenvalue spacings for regular graphs (D. Jakobson, S. D. Miller, I. Rivin, and Z. Rudnick). Pages 317--327 in Emerging Applications of Number Theory (Minneapolis, 1996), The IMA Volumes in Mathematics and its Applications, Vol. 109, Springer, New York, 1999.
[Ka] Continued fraction of cubed roots of primes (S. Kapnick), Junior Thesis, Princeton University, Fall 2002.
[Kar] Applications of heat kernels on Abelian groups: ζ(2n), quadratic reciprocity, Bessel integral (A. Karlsson), preprint.
[KS2] Zeros of zeta functions and symmetries (N. Katz and P. Sarnak), Bull. AMS 36 (1999), 1--26.
[Kob1] Why study equations over finite fields? (N. Koblitz)
[Kob2] Elliptic curve cryptosystems (N. Koblitz), Math. Comp. 48 (1987), no. 177, 203--209.
[KonMi] Benford's Law, Values of L-Functions and the 3x+1 Problem (Alex Kontorovich, Steven J. Miller), Acta Arithmetica 120 (2005), 269--297..
[KonSi] Structure theorem for (d,g,h)-maps (A. Kontorovich and Ya. G. Sinai), Bull. Braz. Math. Soc. (N.S.) 33 (2002), no. 2, 213--224.
[Kua] Digit distribution in the continued fraction of zeta(n) (F. Kuan), Junior Thesis, Princeton University, Fall 2002.
[KR] The distribution of spacings between quadratic residues (P. Kurlberg and Z. Rudnick), Duke Math. J. 100 (1999), no. 2, 211--242.
[Lag1] The 3x+1 problem and its generalizations (J. Lagarias). Pages 305-334 in Organic mathematics (Burnaby, BC, 1995), CMS Conf. Proc., vol. 20, AMS, Providence, RI, 1997.
[Lag2] The 3x+1 problem: An annotated bibliography (J. Lagarias), preprint.
[LaSo] Benford's Law for the 3x+1 function (J. Lagarias and K. Soundararajan), J. London Math. Soc. (2) 74 (2006), no. 2, 289--303. .
[LT] Continued fractions for some algebraic numbers (S. Lang and H. Trotter), J. Reine Angew. Math. 255 (1972), 112--134.
[LP] Gauss, Eisenstein, and the "third" proof of the quadratic reciprocity theorem: Ein kleines Schauspiel (R. Laubenbacher and D. Pengelley), Math. Intelligencer 16 (1994), no. 2, 67--72.
[Law1] Kuzmin's theorem on algebraic numbers (J. Law), Junior Thesis, Princeton University, Fall 2002.
[Law2] The circle method on the binary Goldbach conjecture (J. Law), Junior Thesis, Princeton University, Spring 2003.
[Leh] First order spacings of random matrix eigenvalues (R. Lehman), Junior Thesis, Princeton University, Spring 2000.
[LS] On hats and other covers (H. Lenstra and G. Seroussi), 2002, preprint.
[Le] Sur les lois de probabilite dont dependent les quotients complets et incomplets d'une fraction continue (P. Levy), Bull. Soc. Math. 57 (1929), 178--194.
[LU] Transcendence of e and π (C. Liaw and H. Ulfarsson), notes for Math 252 (Graduate Algebra), Brown University, Spring 2006.
[Lidl] Mathematical aspects of cryptanalysis (R. Lidl). Pages 86--97 in Number Theory and Cryptography (Sydney, 1989), London Mathematical Society Lecture Note Series, vol. 154, Cambridge University Press, Cambridge, 1990.
[Li] Numerical results concerning the distribution of n2 alpha (R. Lipshitz), Junior Thesis, Princeton University, Spring 2000.
[Liu] Statistical behavior of the eigenvalues of random matrices (Y. Liu), Junior Thesis, Princeton University, Spring 2000.
[Mah] Arithmetische Eigenschaften einer Klasse von Dezimalbruchen (K. Mahler), Amsterdam Proc. Konin. Neder. Akad. Wet. 40 (1937), 421--428.
[Mar] Almost modular functions and the distribution of n2 x modulo one (J. Marklof), Int. Math. Res. Not. (2003), no. 39, 2131--2151.
[MaMc] An elliptic curve over Q with rank at least 24 (R. Martin and W. McMillen), Number Theory Listserver, May 2000.
[MMS] Eigenvalue spacing distribution for the ensemble of real symmetric palindromic Toeplitz matrices (Adam Massey, Steven J. Miller and John. Sinsheimer), Journal of Theoretical Probability 20 (2007), no. 3, 637--662.
[Maz1] Modular curves and the Eisenstein ideal (B. Mazur), IHES Publ. Math. 47 (1977), 33--186.
[Maz2] Rational isogenies of prime degree (with an appendix by D. Goldfeld) (B. Mazur), Invent. Math. 44 (1978), no. 2, 129--162.
[Maz3] Number Theory as Gadfly (B. Mazur), Amer. Math. Monthly, 98 (1991), 593--610.
[Maz4] Finding meaning in error terms (B. Mazur), Bull. of the AMS, 45 (2008), no. 2, 185--228.
[McK] The expected eigenvalue distribution of a large regular graph (B. McKay), Linear Algebra Appl. 40 (1981), 203--216.
[McW] The degree sequence of a random graph. I. The models (B. McKay and N. Wormald), Random Structures Algorithms 11 (1997), no. 2, 97--117.
[Meh1] On the statistical properties of level spacings in nuclear spectra (M. Mehta), Nucl. Phys. 18 (1960), 395--419.
[Met] The beginning of the Monte Carlo method (N. Metropolis), Los Alamos Science, No. 15, Special Issue (1987), 125--130.
[MU] The Monte Carlo method (N. Metropolis and S. Ulam), J. Amer. Statist. Assoc. 44 (1949), 335--341.
[Mic1] Independence of the digits of continued fractions (M. Michelini), Junior Thesis, Princeton University, Fall 2002.
[Mic2] Kuzmin's Extraordinaty Zero Measure Set (Matt Michelini).
[Mi1] Various tendencies of non-Poissonian distributions along subsequences of certain transcendental numbers (N. Miller), Junior Thesis, Princeton University, Fall 2002.
[Mi2] Distribution of eigenvalue spacings for band-diagonal matrices (N. Miller), Junior Thesis, Princeton University, Spring 2003.
[Mill] An easier way to show ζ(3) is irrational (Stephen D. Miller).
[Mil1] 1 and 2-Level Densities for Families of Elliptic Curves: Evidence for the Underlying Group Symmetries (Steven J. Miller), Compositio Mathematica 104 (2004), no. 4, 952--992.
[Mil2] Density functions for families of Dirichlet characters (Steven J. Miller).
[Mil3] The Arithmetic Mean and Geometric Inequality (Steven J. Miller).
[Mil4] Differentiating Identities (Steven J. Miller).
[Mil5] The Pythagorean won-loss formula in baseball (Steven J. Miller), Chance Magazine 20 (2007), no. 1, 40--48 (an abridged version appeared in The Newsletter of the SABR Statistical Analysis Committee 16 (February 2006), no. 1, 17--22).
[Mil6] Investigations of zeros near the central point of elliptic curve L-functions (Steven J. Miller), Experimental Mathematics 15 (2006), no. 3, 257--279.
[Mil7] Die battles and order statistics (Steven J. Miller), Class Notes from Math 162: Statistics, Brown University, Spring 2006.
[Mil8] Beyond the Pigeon-Hole Principle: Many pigeons in the same box, Class Notes from Math 162: Statistics, Brown University, Spring 2006.
[MN] Order Statistics and Shifted Almost Benford Behavior (Steven J. Miller and Mark J. Nigrini), preprint.
[M] Use of elliptic curves in cryptography (V. Miller). Pages 417--426 in Advances in cryptology -- CRYPTO '85 (Santa Barbara, CA, 1985), Lecture Notes in Computer Science, Vol. 218, Springer-Verlag, Berlin, 1986.
[Milne] Elliptic Curves (J. S. Milne): online notes.
[Min] Analysis of Benford's law applied to the 3x+1 problem (S. Minteer), Number Theory Working Group, The Ohio State University, 2004.
[Mon1] Primes in arithmetic progression (H. Montgomery), Michigan Math. J. 17 (1970), 33--39.
[Mon2] The pair correlation of zeros of the zeta function (H. Montgomery). Pages 181--193 in Analytic Number Theory, Proceedings of Symposia in Pure Mathematics, vol. 24, AMS, Providence, RI, 1973.
[MS] Beyond pair correlation (H. Montgomery and K. Soundararajan). Pages 507--514 in Paul Erdös and His Mathematics, I (Budapest, 1999), Bolyai Society Mathematical Studies, Vol. 11, János Bolyai Math. Soc., Budapest, 2002.
[MW] Sums of Squares of Integers (C. J. Moreno and S. S. Wagstaff, Jr.), Chapman and Hall, 2006.
[Mu1] Primes in certain arithmetic progressions (R. Murty), Journal of the Madras University, (1988), 161--169.
[NS] Benford's law for linear recurrence sequences (K. Nagasaka and J. S. Shiue), Tsukuba J. Math. 11 (1987), 341--351.
[NT] Continued fraction expansion of 21/3 (J. von Neumann and B. Tuckerman), Math. Tables Aids Comput. 9 (1955), 23--24.
[Ni1] The pentium bug (T. Nicely).
[Ni2] Enumeration to 1014 of the Twin Primes and Brun's Constant (T. Nicely), Virginia J. Sci. 46 (1996), 195--204.
[Nig1] Digital Analysis and the Reduction of Auditor Litigation Risk (M. Nigrini). 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.
[Nig2] The Use of Benford's Law as an Aid in Analytical Procedures (M. Nigrini), Auditing: A Journal of Practice & Theory, 16 (1997), no. 2, 52--67.
[Nov] Asymptotic behavior of the random 3-regular bipartite graph (T. Novikoff), Undergraduate Mathematics Laboratory report, Courant Institute, NYU, 2002.
[Ny] An Application of Diophantine Approximation (J. E. Nymann), The American Mathematical Monthly, 76 (1969), no. 6, 668--671.
[Od2] The 10^22-nd zero of the Riemann zeta function (A. Odlyzko). Pages 139--144 in Proceedings 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.
[Ol] Verification of the Goldbach conjecture up to 6 * 10^16 (T. Oliveira e
Silva), 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/\simtos/goldbach.html.
[Ols] Extremely non-normal continued fractions (L. Olsen), Acta Arith. 108 (2003), no. 2, 191--202.
[Pi] Carmichael numbers up to 1018 (R. G. E. Pinch), preprint.
[Pol] Heuristic reasoning in the theory of numbers (G. Polya), The American Mathematical Monthly, Vol. 66, No. 5. (May, 1959), pp. 375-384.
[vdP1] An introduction to continued fractions (A. van der Poorten). Pages 99-138 in Diophantine Analysis (Kensington, 1985), London Mathematical Society Lecture Note Series, Vol. 109, Cambridge University Press, Cambridge, 1986.
[vdP2] Notes on continued fractions and recurrence sequences (A. van der Poorten). Pages 86--97 in Number theory and cryptography (Sydney, 1989), London Mathematical Society Lecture Note Series, Vol. 154, Cambridge University Press, Cambridge, 1990.
[vdP3] Continued fractions of formal power series (A. van der Poorten). Pages 453--466 in Advances in Number Theory (Kingston, ON, 1991), Oxford Science Publications, Oxford University Press, New York, 1993.
[vdP4] Fractions of the period of the continued fraction expansion of quadratic integers (A. van der Poorten), Bull. Austral. Math. Soc. 44 (1991), no. 1, 155--169.
[vdP5] Continued fraction expansions of values of the exponential function and related fun with continued fractions (A. van der Poorten), Nieuw Arch. Wisk. (4) 14 (1996), no. 2, 221--230.
[PS1] Folded continued fractions (A. van der Poorten and J. Shallit), J. Number Theory 40 (1992), no. 2, 237--250.
[PS2] A specialised continued fraction (A. van der Poorten and J. Shallit), Canad. J. Math. 45 (1993), no. 5, 1067--1079.
[Po] Statistical Theories of Spectra: Fluctuations (C. Porter (editor)), Academic Press, New York, 1965.
[Py] Spacings (R. Pyke), J. Roy. Statist. Soc. Ser. B 27 (1965), 395--449.
[QS1] Rational relation conjectures (R. Qian and D. Steinhauer), Junior Thesis, Princeton University, Fall 2003.
[QS2] Eigenvalues of weighted random graphs (R. Qian and D. Steinhauer), Junior Thesis, Princeton University, Spring 2003.
[Rai] The first digit problem (R. A. Raimi), Amer. Math. Monthly 83 (1976), no. 7, 521--538.
[Ric] An investigation of expanders and ramanujan graphs along random walks of cubic bipartite graphs (P. Richter), Junior Thesis, Princeton University, Spring 2001.
[RDM] Continued fraction of algebraic numbers (R. D. Richtmyer, M. Devaney, and N. Metropolis), Numer. Math. 4 (1962), 68--84.
[Rie] On the sign of the difference π(x) - Li(x) (H. J. J. te Riele), Mathematics of Computation, Vol. 48, No. 177. (Jan., 1987), pp. 323-328.
[Ri] Uber die Anzahl der Primzahlen unter einer gegebenen Grosse (G. F. B. Riemann), Monatsber. Konigl. Preuss. Akad. Wiss. Berlin, Nov. 1859, 671--680.
[RSA] A method for obtaining digital signatures and public key cryptosystems (R. Rivest, A. Shamir, and L. Adleman), Comm. ACM 21 (1978), 120--126.
[Ro] Rational approximations to algebraic numbers (K. Roth), Mathematika 2 (1955), 1--20.
[Rub1] A simple heuristic proof of Hardy and Littlewood's conjecture B (M. Rubinstein), Amer. Math. Monthly 100 (1993), no. 5, 456--460.
[Rub2] Low-lying zeros of L-functions and random matrix theory (M. Rubinstein), Duke Math. J. 109 (2001), no. 1, 147--181.
[RubSa] Chebyshev's bias (M. Rubinstein and P. Sarnak), Experiment. Math. 3 (1994), no. 3, 173--197.
[RS] Zeros of principal L-functions and random matrix theory (Z. Rudnick and P. Sarnak), Duke J. of Math. 81 (1996), 269--322.
[RS2] The pair correlation function of fractional parts of polynomials (Z. Rudnick and P. Sarnak), Comm. Math. Phys. 194 (1998), no. 1, 61--70.
[RSZ] The distribution of spacings between the fractional parts of n2 alpha (Z. Rudnick, P. Sarnak, and A. Zaharescu), Invent. Math.145 (2001), no. 1, 37--57.
[RZ1] A metric result on the pair correlation of fractional parts of sequences (Z. Rudnick and A. Zaharescu), Acta Arith. 89 (1999), no. 3, 283--293.
[RZ2] The distribution of spacings between fractional parts of lacunary sequences (Z. Rudnick and A. Zaharescu), Forum Math. 14 (2002), no. 5, 691--712.
[Sai] A new proof of Euclid's Theorem (F. Saidak), Amer. Math. Monthly 113 (2006), no. 10, 937--938.
[Sch] Prime Spacing and the Hardy-Littlewood Conjecture B (D. Schmidt), Junior Thesis, Princeton University, Spring 2001.
[Si] The Classical Moment Problem as a Self-Adjoint Finite Difference Operator (Barry Simon).
[SM] Eigenvalue density of correlated complex random Wishart matrices (S. Simon and A. Moustakas), Bell Labs Technical Memo, 2004.
[Sk] On the difference π(x) - Li(x) (S. Skewes), J. London Math. Soc. 8 (1933), 277--283.
[Sl] On-Line Encyclopedia of Integer Sequences (N. Sloane).
[Sn] Derivatives of random matrix characteristic polynomials with applications to elliptic curves (N. Snaith), J. Phys. A 38 (2005), no. 48, 10345--10360.
[So] Small gaps between prime numbers: The work of Goldston-Pintz-Yildirim (K. Soundararajan), Bull. of the AMS 44 (2007), no. 1, 1--18.
[Str] The Banach-Tarski paradox (K. Stromberg), Amer. Math. Monthly 86 (1979), no. 3, 151--161.
[Sz] On the length of continued fractions representing a rational number with given denominator (P. Szusz), Acta Arithmetica 37 (1980), 55--59.
[Ta] The Gamma function and Kuzmin's theorem (C. Taylor), Junior Thesis, Princeton University, Fall 2002.
[TW] Ring-theoretic properties of certain Hecke algebras (R. Taylor and A. Wiles), Ann. Math. 141 (1995), 553--572.
[TrWi] Correlation functions, cluster functions, and spacing distributions for random matrices (C. Tracy and H. Widom), J. Statist. Phys., 92 (1998), no. 5--6, 809--835.
[Va] On a variance associated with the distribution of primes in arithmetic progression (R. C. Vaughan), Proc. London Math. Soc. (3) 82 (2001), 533--553.
[VW] Waring's problem: a survey. Pages 301--340 in Number Theory for the Millennium, III (R. C. Vaughan and T. D. Wooley) (Urbana, IL, 2000), A. K. Peters, Natick, MA, 2002.
[Vin1] Representation of an odd number as the sum of three primes (I. Vinogradov), Doklady Akad. Nauk SSSR, 15 (1937), no. 6--7, 291--294.
[Vin2] Some theorems concerning the theory of primes (I. Vinogradov), Mat. Sbornik, 2 (1937), no. 44, 179--195.
[Vo] A sharpening of Li's criterion for the Riemann hypothesis (A. Voros), preprint.
[Wed] ZetaGrid (S. Wedeniwski).
[Wei1] Numbers of Solutions of Equations in Finite Fields (A. Weil), Bull. Amer. Math. Soc. 14 (1949), 497--508.
[Wei2] Prehistory of the zeta-function (A. Weil). Pages 1--9 in Number Theory, Trace Formulas and Discrete Groups (Oslo, 1987), Academic Press, Boston, 1989.
[Weir] The local behavior of Germain primes (B. Weir), Undergraduate Mathematics Laboratory report, Courant Institute, NYU, 2002.
[We] MathWorld --- A Wolfram Web Resource (E. Weisstein).
[Wig2] Characteristic vectors of bordered matrices with infinite dimensions (E. Wigner), Ann. of Math. 2 (1955), no. 62, 548--564.
[Wig4] Characteristic vectors of bordered matrices with infinite dimensions. II (E. Wigner), Ann. of Math. Ser. 2 65 (1957), 203--207.
[Wig5] On the distribution of the roots of certain symmetric matrices (E. Wigner), Ann. of Math. Ser. 2 67 (1958), 325--327.
[Wi] Modular elliptic curves and Fermat's last theorem (A. Wiles), Ann. Math. 141 (1995), 443--551.
[Wir] On the theorem of Gauss-Kuzmin-Levy and a Frobenius-type theorem for function spaces (E. Wirsing), Acta Arith. 24 (1974) 507--528.
[Wis] The generalized product moment distribution in samples from a normal multivariate population (J. Wishart), Biometrika 20 A (1928), 32--52.
[Wor] Models of random regular graphs (N. C. Wormald). Pages 239--298 in Surveys in combinatorics, 1999 (Canterbury) London Mathematical Society Lecture Note Series, vol. 267, Cambridge University Press, Cambridge, 1999.
[Wo] Large improvements in Waring's problem (T. Wooley), Ann. of Math. (2), 135 (1992), no. 1, 131--164.
[Za] A Generalization of Wigner's Law (Inna Zakharevich),Comm. Math. Phys. 268 (2006), no. 2, 403--414.
[Zu] One of the numbers zeta(5), zeta(7), zeta(9), zeta(11) is irrational (W. Zudilin), Uspekhi Mat. Nauk 56 (2001), 149-150.