• nkα mod 1 and Poissonian behavior
    • General theory
      • An Introduction to the Moment Problem (Jacob Christiansen).
    • Examples
      • Poisson statistics via the Chinese remainder theorem (A. Granville and P. Kurlberg), preprint.
      • The distribution of spacings between quadratic residues (P. Kurlberg and Z. Rudnick), Duke Math. J. 100 (1999), no. 2, 211--242.
    • Student Projects
      • Numerical results concerning the distribution of n2 alpha (R. Lipshitz), Junior Thesis, Princeton University, Spring 2000.
      • Prime Spacing and the Hardy-Littlewood Conjecture B (D. Schmidt), Junior Thesis, Princeton University, Spring 2001.

 

  • Random Matrix Theory
    • Historical overview and surveys
      • The spectrum of Riemannium (B. Hayes), American Scientist 91 (2003), no. 4, 296--300.
      • Developments in Random Matrix Theory (P. J. Forrester, N. C. Snaith, and J. J. M. Verbaarschot). In Random matrix theory, J. Phys. A 36 (2003), no. 12, R1--R10.
      • Random matrices and L-functions (J. P. Keating and N. C. Snaith). In Random matrix theory, J. Phys. A 36 (2003), no. 12, 2859--2881
    • General theory
      • An Introduction to the Moment Problem (Jacob Christiansen).
      • Some elementary results around the Wigner semicircle law (A. Boutet de Monvel and A. Khorunzhy), lecture notes.
      • Log-gases and Random matrices (Peter Forrester):  book in progress
    • Eigenvalue distribution of special ensembles of matrices
      • 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
      • Distribution 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.
      • 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.
      • A Generalization of Wigner's Law (Inna Zakharevich),Comm. Math. Phys. 268 (2006), no. 2, 403--414.
    • d-Regular graphs
      • 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.
      • The expected eigenvalue distribution of a large regular graph (B. McKay), Linear Algebra Appl. 40 (1981), 203--216.
      • The distribution of the second largest eigenvalue in families of random regular graphs (S. J. Miller, T. Novikoff and A. Sabelli), Experimental Mathematics 17 (2008), no. 2, 231--244.
      • 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.
    • Student Projects
      • First order spacings of random matrix eigenvalues (R. Lehman), Junior Thesis, Princeton University, Spring 2000.
      • Statistical behavior of the eigenvalues of random matrices (Y. Liu), Junior Thesis, Princeton University, Spring 2000.
      • Distribution of eigenvalue spacings for band-diagonal matrices (N. Miller), Junior Thesis, Princeton University, Spring 2003.
      • Eigenvalues of weighted random graphs (R. Qian and D. Steinhauer), Junior Thesis, Princeton University, Spring 2003.
    • Classic papers
      • Statistical theory of the energy levels of complex systems: I, II, III (F. Dyson), J. Mathematical Phys., 3 (1962) 140--156, 157--165, 166--175.
      • The threefold way. Algebraic structure of symmetry groups and ensembles in quantum mechanics (F. Dyson), J. Mathematical Phys., 3 (1962) 1199--1215.

 

  • The Circle Method
    • Waring's Problem
      • 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.
      • Waring's problem (J. Cisneros), Junior Thesis, Princeton University, Spring 2001.
    • Goldbach's Problem
      • The circle method on the binary Goldbach conjecture (J. Law), Junior Thesis, Princeton University, Spring 2003.
      • 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.
    • Germain Primes
      • The local behavior of Germain primes (B. Weir), Undergraduate Mathematics Laboratory report, Courant Institute, NYU, 2002.

 

  • Benford's Law
    • General Theory

      • The first-digit phenomenon (T. Hill), American Scientist 86 (1996), 358--363.

      • A statistical derivation of the significant-digit law (T. Hill), Statistical Science 10 (1996), 354--363.

      • The first digit problem (R. A. Raimi), Amer. Math. Monthly 83 (1976), no. 7, 521--538.

    • Applications
      • I've got your number (M. Nigrini), Journal of Accountacy (1999).
    • Examples
      • Benford's Law, Values of L-Functions and the 3x+1 Problem (Alex Kontorovich, Steven J. Miller), Acta Arithmetica 120 (2005), 269--297.
      • Benford's Law for the 3x+1 function (J. Lagarias and K. Soundararajan),  J. London Math. Soc. (2) 74 (2006), no. 2, 289--303.
      • Order Statistics and Shifted Almost Benford Behavior (Steven J. Miller and Mark J. Nigrini),  International Journal of Mathematics and Mathematical Sciences, Volume 2008 (2008), Article ID 382948, 19 pages, doi:10.1155/2008/382948.
    • Numerics
      • Analysis of Benford's law applied to the 3x+1 problem (S. Minteer), Number Theory Working Group, The Ohio State University, 2004.

 

  • Riemann Zeta Function and L-functions
    • Historical overview
      • The spectrum of Riemannium (B. Hayes), American Scientist 91 (2003), no. 4, 296--300.
    • General surveys
      • L-Functions and random matrices (J. B. Conrey). Pages 331--352 in Mathematics unlimited --- 2001 and Beyond, Springer-Verlag, Berlin, 2001.
      • The Riemann hypothesis (J. B. Conrey), Notices of the AMS, 50 (2003), no. 3, 341--353.
      • Zeros of zeta functions and symmetries (N. Katz and P. Sarnak), Bull. AMS 36 (1999), 1--26.
      • Random matrices and L-functions (J. P. Keating and N. C. Snaith). In Random matrix theory, J. Phys. A 36 (2003), no. 12, 2859--2881
    • Numerical computations
      • Investigations of zeros near the central point of elliptic curve L-functions (Steven J. Miller), Experimental Mathematics 15 (2006), no. 3, 257--279.
      • 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.
      • The 1013 first zeros of the Riemann zeta function, and zeros computation at very large height, A. Odlyzko, preprint.
      • On the distribution of spacings between zeros of the zeta function (A. Odlyzko), Math. Comp. 48 (1987), no. 177, 273--308.
      • 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.

       

  • More Sums Than Differences
    • Many sets have more sums than differences (Martin and O'Bryant)
    • Problems in Additive Number Theory I (Nathanson)
    • When almost all sets are difference dominated (Steven Miller and Peter Hegarty)
    •

 

  • The 3x+1 Problem
    • 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.
    • The 3x+1 problem: An annotated bibliography (J. Lagarias), preprint.
    • Benford's Law, Values of L-Functions and the 3x+1 Problem (Alex Kontorovich, Steven J. Miller), Acta Arithmetica 120 (2005), 269--297.
    • Benford's Law for the 3x+1 function (J. Lagarias and K. Soundararajan),  J. London Math. Soc. (2) 74 (2006), no. 2, 289--303.