Skip to main content
SpringerLink
Log in
Book cover

Open Problems in Mathematics pp 123–171Cite as

From Quantum Systems to L-Functions: Pair Correlation Statistics and Beyond

  • Chapter
  • First Online:

Abstract

The discovery of connections between the distribution of energy levels of heavy nuclei and spacings between prime numbers has been one of the most surprising and fruitful observations in the twentieth century. The connection between the two areas was first observed through Montgomery’s work on the pair correlation of zeros of the Riemann zeta function. As its generalizations and consequences have motivated much of the following work, and to this day remains one of the most important outstanding conjectures in the field, it occupies a central role in our discussion below. We describe some of the many techniques and results from the past sixty years, especially the important roles played by numerical and experimental investigations, that led to the discovery of the connections and progress towards understanding the behaviors. In our survey of these two areas, we describe the common mathematics that explains the remarkable universality. We conclude with some thoughts on what might lie ahead in the pair correlation of zeros of the zeta function, and other similar quantities.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   149.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   199.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    A total neutron cross section is defined as

    $$\displaystyle\begin{array}{rcl} \frac{\mathrm{Number\ of\ events\ of\ all\ types\ per\ unit\ time\ per\ nucleus}} {\mathrm{Number\ of\ incident\ neutrons\ per\ unit\ time\ per\ unit\ area}},& & {}\\ \end{array}$$

    and has the dimensions of area (the standard unit is the barn, 10−24 cm2).

  2. 2.

    The width, Γ, is related to the lifetime, τ, by the uncertainty relation Γ = h∕2π τ, where h is Planck’s constant. The finite width (lack of energy definition) is due to the fact that a resonant state can decay by emitting a particle, or radiation, whereas a state of definite energy must be a stationary state.

  3. 3.

    If v is an eigenvector with eigenvalue \(\lambda\) of a Hermitian matrix A (so A = A ∗ with A ∗ the complex conjugate transpose of A, then v ∗(Av) = v ∗(A ∗ v) = (Av)∗ v); the first expression is \(\lambda \vert \vert v\vert \vert ^{2}\) while the last is \(\overline{\lambda }\vert \vert v\vert \vert ^{2}\), with \(\vert \vert v\vert \vert ^{2} = v^{{\ast}}v =\sum \vert v_{i}\vert ^{2}\) non-zero. Thus \(\lambda = \overline{\lambda }\), and the eigenvalues are real. This is one of the most important properties of Hermitian matrices, as it allows us to order the eigenvalues.

  4. 4.

    In fact, one of the authors has used Weibull distributions to model run production in major league baseball, giving a theoretical justification for Bill James’ Pythagorean Won-Loss formula [103].

  5. 5.

    Obviously this Weibull cannot be a normal distribution, as they have very different decay rates for large x, and this Weibull is a one-sided distribution! What we mean is that for 0 ≤ x ≤ 2 this Weibull is well approximated by a normal distribution which shares its mean and variance, which are (respectively) Γ(4∕3) ≈ 0. 893 and Γ(5∕3) −Γ(4∕3)2 ≈ 0. 105.

  6. 6.

    In other words, while the above is enough to prove that the class number tends to infinity, we cannot use that argument to produce an explicit constant Q n for each n so that we could assert that the class number is at least n if q ≥ Q n . One of the best illustrations of the importance of effective constants is the following joke: There is a constant T 0 such that if all the non-trivial zeros of ζ(s) in the critical strip up to height T 0 are on the critical line, then they all are and the Riemann Hypothesis is true; in other words, it suffices to check up to a finite height! To see this, if the Riemann Hypothesis is true we may take T 0 to be 0, while if it is false we take T 0 to be 1 more than the height of the first exemption. We have therefore shown a constant exists, but such information is completely useless!

References

  1. Y. Alhassid, The Statistical Theory of Quantum Dots, Rev. Mod. Phys. 72 (2000), 895–968.

    Article  Google Scholar 

  2. L. Alpoge, N. Amersi, G. Iyer, O. Lazarev, S. J. Miller and L. Zhang, Maass waveforms and low-lying zeros, in Analytic Number Theory: In Honor of Helmut Maier’s 60th Birthday, Springer-Verlag, 2015.

    MATH  Google Scholar 

  3. L. Alpoge and S. J. Miller, The low-lying zeros of level 1 Maass forms, Int. Math. Res. Not. IMRN 2010, no. 13, 2367–2393.

    Google Scholar 

  4. S. Arno, The imaginary quadratic fields of class number 4, Acta. Arith. 60 (1992), no. 4, 321–334.

    MathSciNet  MATH  Google Scholar 

  5. S. Arno, M. Robinson, and F. Wheeler, Imaginary quadratic fields with small odd class number, Acta. Arith. 83 (1998), no. 4, 296–330.

    MathSciNet  MATH  Google Scholar 

  6. N. Austern, Fast Neutron Physics (Vol. 2), Interscience, 1963.

    Google Scholar 

  7. J. Baik, A. Borodin, P. Deiftn and T Suidan, A Model for the Bus System in Cuernevaca (Mexico), J. Phys. A: Math. Gen. 39 (2006) 8965–8975. http://arxiv.org/abs/math/0510414.

    Article  MATH  Google Scholar 

  8. A. Baker, Linear forms in the logatirhms of algebraic numbers, Mathematika 13 (1966) 204–216.

    Article  MATH  Google Scholar 

  9. A. Baker, Imaginary quadratic fields with class number two, Ann. of Math. 2 (1971) 139–152.

    Article  MATH  Google Scholar 

  10. H. A. Bethe, Nuclear Physics, B. Nuclear Dynamics, Theoretical, Rev. Mod. Phys. 9 (1937), 69–249.

    Article  MATH  Google Scholar 

  11. J. M. Blatt and V. F. Weisskopf, Theoretical Nuclear Physics, Wiley, 1952.

    MATH  Google Scholar 

  12. E. Bogomolny, O. Bohigas, P. Leboeuf, and A.G. Monastra, On the spacing distribution of the Riemann zeros: corrections to the asymptotic result, J. Phys. A: Math. Gen. 39 (2006), 10743–10754.

    Article  MathSciNet  MATH  Google Scholar 

  13. E. Bogomolny and J.P. Keating, Gutzwiller’s trace formula and spectral statistics: beyond the diagonal approximation, Phys. Rev. Lett. 77 (1996), 1472–1475.

    Article  MATH  Google Scholar 

  14. A. Bohr and B. Mottelson, Dan. Nat. Fys. Medd. 27 (1953), no. 16.

    Google Scholar 

  15. N. Bohr, Neutron Capture and Nuclear Constitution, Nature 137 (1936), 344–348.

    Article  MATH  Google Scholar 

  16. O. Bohigas, M. Giannoni and C. Schmit, Characterization of chaotic quantum spectra and universality of level fluctuation laws, Phys. Rev. Lett. 52 (1984), 1–4.

    Article  MathSciNet  MATH  Google Scholar 

  17. P. Borwein, S. Choi and B. Rooney, eds. The Riemann Hypothesis: A Resource for the Afficionado and Virtuoso Alike, CMS Books in Mathematics, Springer, New York, 2008.

    MATH  Google Scholar 

  18. 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.

    Article  MathSciNet  Google Scholar 

  19. Clay Mathematics Institute, webpage on the Riemann Hypothesis, http://www.claymath.org/millenium-problems/riemann-hypothesis.

  20. J. B. Conrey, L-Functions and random matrices. Pages 331–352 in Mathematics unlimited — 2001 and Beyond, Springer-Verlag, Berlin, 2001.

    Google Scholar 

  21. J. B. Conrey, The Riemann hypothesis, Notices of the AMS, 50 (2003), no. 3, 341–353.

    MathSciNet  MATH  Google Scholar 

  22. B. Conrey, D. Farmer, P. Keating, M. Rubinstein and N. Snaith, Integral moments of L-functions, Proc. London Math. Soc. (3) 91 (2005), no. 1, 33–104.

    Google Scholar 

  23. J. B. Conrey, D. W. Farmer and M. R. Zirnbauer, Autocorrelation of ratios of L-functions, Comm. Number Theor. Phys. 2 (2008), no. 3, 593–636

    Article  MathSciNet  MATH  Google Scholar 

  24. J. B. Conrey, D. W. Farmer and M. R. Zirnbauer, Howe pairs, supersymmetry, and ratios of random characteristic polynomials for the classical compact groups, preprint. http://arxiv.org/abs/math-ph/0511024.

  25. J. B. Conrey and H. Iwaniec, Spacing of Zeros of Hecke L-Functions and the Class Number Problem, Acta Arith. 103 (2002) no. 3, 259–312.

    Article  MathSciNet  MATH  Google Scholar 

  26. J. B. Conrey and N. C. Snaith, Applications of the L-functions Ratios Conjecture, Proc. London Math. Soc. (3) 94 (2007), no. 3, 594–646.

    Google Scholar 

  27. H. Davenport, Multiplicative Number Theory, 2nd edition, Graduate Texts in Mathematics 74, Springer-Verlag, New York, 1980, revised by H. Montgomery.

    Google Scholar 

  28. C. J. de la Vallée Poussin, Recherches analytiques la théorie des nombres premiers, Ann. Soc. scient. Bruxelles 20 (1896), 183–256. Reprinted in [17].

    Google Scholar 

  29. H. Derrien, L. Leal, and N. Larson, Status of new evaluation of the neutron resonance parameters of  238U at ORNL, PHYSOR 2004, Available on CD-ROM. Amer. Nucl. Soc. LaGrange Park, IL.

    Google Scholar 

  30. J. S. Desjardins, J. Rosen, J. Rainwater and W. Havens Jr., Slow neutron resonance spectroscopy II, Phys. Rev. 120 (1960) 2214–2224.

    Article  Google Scholar 

  31. Persi Diaconis, “What is a random matrix?”, Notices of the Amer. Math. Soc. 52 (2005) 1348–1349.

    MathSciNet  MATH  Google Scholar 

  32. Persi Diaconis, Patterns of Eigenvalues: the 70 th Josiah Willard Gibbs Lecture, Bull. Amer. Math. Soc. 40 (2003) 155–178.

    Google Scholar 

  33. E. Dueñez, D. K. Huynh, J. C. Keating, S. J. Miller and N. Snaith, The lowest eigenvalue of Jacobi Random Matrix Ensembles and Painlevé VI, Journal of Physics A: Mathematical and Theoretical 43 (2010) 405204 (27pp).

    Article  MathSciNet  MATH  Google Scholar 

  34. E. Dueñez, D. K. Huynh, J. C. Keating, S. J. Miller and N. Snaith, Models for zeros at the central point in families of elliptic curves (with Eduardo Dueñez, Duc Khiem Huynh, Jon Keating and Nina Snaith), J. Phys. A: Math. Theor. 45 (2012) 115207 (32pp).

    Article  MATH  Google Scholar 

  35. E. Dueñez and S. J. Miller, The low lying zeros of a GL(4) and a GL(6) family of L-functions, Compositio Mathematica 142 (2006), no. 6, 1403–1425.

    Article  MathSciNet  MATH  Google Scholar 

  36. E. Dueñez and S. J. Miller, The effect of convolving families of L-functions on the underlying group symmetries, Proceedings of the London Mathematical Society, 2009; doi: 10.1112/plms/pdp018.

    MATH  Google Scholar 

  37. 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.

    Google Scholar 

  38. F. Dyson, The threefold way. Algebraic structure of symmetry groups and ensembles in quantum mechanics, J. Mathematical Phys., 3 (1962) 1199–1215.

    Article  MathSciNet  MATH  Google Scholar 

  39. H. M. Edwards, Riemann’s Zeta Function, Academic Press, New York, 1974.

    MATH  Google Scholar 

  40. A. Entin, E. Roditty-Gershon and Z. Rudnick, Low-lying zeros of quadratic Dirichlet L-functions, hyper-elliptic curves and Random Matrix Theory, Geometric and Functional Analysis 23 (2013), no. 4, 1230–1261.

    Article  MathSciNet  MATH  Google Scholar 

  41. L. Erdös, J. A. Ramirez, B. Schlein and H.-T. Yau, Bulk Universality for Wigner Matrices, Comm. Pure Appl. Math. 63 (2010), no. 7, 895–925

    MathSciNet  MATH  Google Scholar 

  42. L. Erdös, B. Schlein and H.-T. Yau, Wegner estimate and level repulsion for Wigner random matrices, Int. Math. Res. Not. IMRN (2010), no. 3, 436–479

    Google Scholar 

  43. P. Erdös, Démonstration élémentaire du théor \(\grave{\mathrm{e}}\) me sur la distribution des nombres premiers, Scriptum 1, Centre Math\(\grave{\mathrm{e}}\) matique, Amsterdam, 1949.

    Google Scholar 

  44. E. Fermi and E. Amaldi, La Ricercio Scientifica 6 (1935), 544.

    Google Scholar 

  45. H. Feshbach, C. E. Porter, and V. F. Weisskopf, Model for Nuclear Reactions with Neutrons, Phys. Rev. 96 (1954), 448–464.

    Article  MATH  Google Scholar 

  46. D. Fiorilli and S. J. Miller, Surpassing the Ratios Conjecture in the 1-level density of Dirichlet L-functions, Algebra & Number Theory Vol. 9 (2015), No. 1, 13–52.

    Google Scholar 

  47. F. Firk, Neutron Time-of-Flight Spectrometers in Detectors in Nuclear Science (editor D. Allan Bromley, North-Holland, Amsterdam (1979)), special issue Nucl. Instr. and Methods, 162 (1979), 539–563.

    Google Scholar 

  48. F. Firk, J. E. Lynn and M. Moxon, Parameters of neutron resonances inU238 below 1.8 keV, Proc. Kingston Intern. Conf. on Nuclear Structure, University of Toronto Press, Toronto (1960), 757–759.

    Google Scholar 

  49. F. Firk and E. Melkonian, Total Neutron Cross Section Measurements in Experimental Neutron Resonance Spectroscopy (editor, J. A. Harvey), Academic Press, New York (1970), 101–154.

    Google Scholar 

  50. F. W. K. Firk and S. J. Miller, Nuclei, Primes and the Random Matrix Connection, Symmetry 1 (2009), 64–105; doi:10.3390/sym1010064. http://www.mdpi.com/2073-8994/1/1/64.

    Article  MathSciNet  Google Scholar 

  51. F. Firk, G. Reid and J. Gallagher, High resolution neutron time- of-flight experiments using the Harwell 15 MeV linear electron accelerator, Nucl. Instr. 3 (1958), 309–315.

    Article  Google Scholar 

  52. P. Forrester, Log-gases and random matrices, London Mathematical Society Monographs 34, Princeton University Press, Princeton, NJ 010.

    Google Scholar 

  53. E. Fouvry and H. Iwaniec, Low-lying zeros of dihedral L-functions, Duke Math. J. 116 (2003), no. 2, 189–217.

    Article  MathSciNet  MATH  Google Scholar 

  54. J. French, V. Kota, A. Pandey and S. Tomosovic, Bounds on time-reversal non-invariance in the nuclear Hamiltonian, Phys. Rev. Lett. 54 (1985), 2313–2316.

    Article  Google Scholar 

  55. P. Gao, N-level density of the low-lying zeros of quadratic Dirichlet L-functions, Ph. D thesis, University of Michigan, 2005.

    Google Scholar 

  56. J. Garg, J. Rainwater, J. Peterson and W. Havens Jr., Neutron resonance spectroscopy III.Th232 and U238, Phys. Rev. 134 (1964) B985–1009.

    Google Scholar 

  57. M. Gaudin, Sur la loi limite de l’espacement des valeurs propres d’une matrice aléatoire, Nucl. Phys. 25 (1961) 447–458.

    Article  MATH  Google Scholar 

  58. C. Gauss, Disquisitiones Arithmeticæ, (1801)

    MATH  Google Scholar 

  59. 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.

    MathSciNet  MATH  Google Scholar 

  60. D. Goldfeld, The conjectures of Birch and Swinnerton-Dyer and the class number of quadratic fields, Journées Arith. De Caen (Univ. Caen, Caen, 1976), Asteérisque nos. 41–42, Soc. Math. France, (1977) 219–227.

    Google Scholar 

  61. D. Goldfeld, Gauss’s class number problem for imaginary quadratic fields, Bull. Amer. Math. Soc. (N.S.) 13 (1985), no. 1, 23–37.

    Google Scholar 

  62. D. Goldfeld, The Elementary proof of the Prime Number Theorem, An Historical Perspective. Pages 179–192 in Number Theory, New York Seminar 2003, eds. D. and G. Chudnovsky, M. Nathanson, Springer-Verlag, New York, 2004.

    Google Scholar 

  63. D. Goldfeld and A. Kontorovich, On theGL (3) Kuznetsov formula with applications to symmetry types of families of L-functions, In Automorphic Representations and L-Functions (ed. D. Prasad et al), Tata Institute (2013), 263–310.

    Google Scholar 

  64. D. A, Goldston, Notes on pair correlation of zeros and prime numbers, Notes on pair correlation of zeros and prime numbers, in Recent perspectives in random matrix theory and number theory, 79–110, London Math. Soc. Lecture Note Ser., 322, Cambridge Univ. Press, Cambridge, 2005. http://arxiv.org/pdf/math/0412313v1.

  65. S. M. Gonek, C. Hughes and J. P. Keating, A hybrid Euler-Hadamard product for the Riemann zeta function, Duke Math. J. 136 (2007) 507–549.

    MathSciNet  MATH  Google Scholar 

  66. B. Gross and D. Zagier, Heegner points and derivatives of L-series, Invent. Math. 84 (1986), no. 2, 225–320.

    Article  MathSciNet  MATH  Google Scholar 

  67. A. Güloğlu, Low Lying Zeros of Symmetric Power L-Functions, Int. Math. Res. Not. (2005), no. 9, 517–550.

    Google Scholar 

  68. I. I. Gurevich and M. I. Pevzner, Repulsion of nuclear levels, Physica 22 (1956), 1132.

    Article  Google Scholar 

  69. J. Hadamard, Sur la distribution des zéros de la fonction ζ(s) et ses conséquences arithmétiques, Bull. Soc. math. France 24 (1896), 199–220. Reprinted in [17].

    Google Scholar 

  70. G. H. Hardy and E. Wright, An Introduction to the Theory of Numbers, 5th edition, Oxford Science Publications, Clarendon Press, Oxford, 1995.

    MATH  Google Scholar 

  71. J. Harvey and D. Hughes, Spacings of nuclear energy levels, Phys. Rev. 109 (1958), 471–479.

    Article  Google Scholar 

  72. R. Haq, A. Pandey and O. Bohigas, Fluctuation properties of nuclear energy levels: do theory and experiment agree? Phys. Rev. Lett. 48 (1982), 1086–1089.

    Article  Google Scholar 

  73. B. Hayes, The spectrum of Riemannium, American Scientist 91 (2003), no. 4, 296–300.

    Article  Google Scholar 

  74. K. Heegner, Diophantische Analysis und Modulfunktionen, Math. Z. 56 (1952) 227–253.

    Article  MathSciNet  MATH  Google Scholar 

  75. H. Heilbronn, On the class number in imaginary quadratic fields, Quart. J. Math. Oxford Ser. 2 5 (1934) 150–160.

    Article  MATH  Google Scholar 

  76. D. Hejhal, On the triple correlation of zeros of the zeta function, Internat. Math. Res. Notices 1994, no. 7, 294–302.

    Google Scholar 

  77. D. A. Hejhal and A. M. Odlyzko, Alan Turing and the Riemann zeta function, Alan Turing – His Work and Impact (J. van Leeuwen and S.B. Cooper, eds.), Elsevier Science, 2012

    Google Scholar 

  78. D. Hughes, Neutron Cross Sections, Pergamon Press, New York (1957).

    Google Scholar 

  79. C. Hughes and S. J. Miller, Low-lying zeros of L-functions with orthogonal symmetry, Duke Math. J. 136 (2007), no. 1, 115–172.

    Google Scholar 

  80. C. Hughes and Z. Rudnick, Linear statistics of low-lying zeros of L-functions, Quart. J. Math. Oxford 54 (2003), 309–333.

    Article  MathSciNet  MATH  Google Scholar 

  81. D. J. Hughes and R. B. Schwartz, Brookhaven National Laboratory Report, No. BNL–325 (1958).

    Google Scholar 

  82. H. Iwaniec and E. Kowalski, Analytic Number Theory, AMS Colloquium Publications, Vol. 53, AMS, Providence, RI, 2004.

    MATH  Google Scholar 

  83. 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.

    Article  MathSciNet  MATH  Google Scholar 

  84. N. Katz and P. Sarnak, Random Matrices, Frobenius Eigenvalues and Monodromy, AMS Colloquium Publications 45, AMS, Providence, 1999.

    Google Scholar 

  85. N. Katz and P. Sarnak, Zeros of zeta functions and symmetries, Bull. AMS 36 (1999), 1–26.

    Article  MathSciNet  MATH  Google Scholar 

  86. J. P. Keating and N. C. Snaith, Random matrices and L-functions, Random matrix theory, J. Phys. A 36 (2003), no. 12, 2859–2881.

    Article  MathSciNet  MATH  Google Scholar 

  87. M. Krbalek and P. Seba, The statistical properties of the city transport in Cuernavaca (Mexico) and Random matrix ensembles, J. Phys. A 33 (2000), 229–234.

    Article  MATH  Google Scholar 

  88. L. Landau and Ya. Smorodinski, Lektsii po teori atomnogo yadra, Gos Izd. tex- teoreyicheskoi Lit. Moscow, (1955) 92–93.

    Google Scholar 

  89. A. M. Lane, Theory of radiative capture reactions, Nucl. Phys. 11 (1959), 625–645.

    Article  Google Scholar 

  90. A. M. Lane and J. P. Elliott, Handbuch der Physik (Vol. 39), Springer-Verlag, 1957.

    Google Scholar 

  91. A. M. Lane and J. E. Lynn, Analysis of experimental data on nucleon capture reactions, Nucl. Phys. 11 (1959), 646–645.

    Article  Google Scholar 

  92. A. M. Lane, R. G. Thomas and E. P. Wigner, Giant Resonance Interpretation of the Nucleon-Nucleus Interaction, Phys. Rev. 98 (1955), 693–701.

    Article  Google Scholar 

  93. D. H. Lehmer, E. Lehmer, and D. Shanks, Integer sequences having prescribed quadratic character, Math. Comp. 24 (1970), 433–451

    MathSciNet  MATH  Google Scholar 

  94. J. Levinson and S. J. Miller, The n-level density of zeros of quadratic Dirichlet L-functions, Acta Arithmetica 161 (2013), 145–182.

    Article  MathSciNet  MATH  Google Scholar 

  95. J. E. Lynn, The Theory of Neutron Resonance Reactions, The Clarendon Press, Oxford (1968).

    Google Scholar 

  96. B. Mackall, S. J. Miller, C. Rapti and K. Winsor, Lower-Order Biases in Elliptic Curve Fourier Coefficients in Families, to appear in the Conference Proceedings of the Workshop on Frobenius distributions of curves at CIRM in February 2014.

    Google Scholar 

  97. M. G. Mayer, On Closed Shells in Nuclei. II, Phys. Rev. 75 (1949), 1969–1970.

    Article  Google Scholar 

  98. M. L. Mehta, On the statistical properties of level spacings in nuclear spectra, Nucl. Phys. 18 (1960), 395–419.

    Article  MathSciNet  MATH  Google Scholar 

  99. M. L. Mehta, Random Matrices, 3rd edition, Elsevier, San Diego, CA (2004)

    MATH  Google Scholar 

  100. M. L. Mehta and M. Gaudin, On the density of the eigenvalues of a random matrix, Nuclear Physics 18 (1960), 420–427.

    Article  MathSciNet  MATH  Google Scholar 

  101. S. J. Miller, 1- and 2-level densities for families of elliptic curves: evidence for the underlying group symmetries, Compositio Mathematica 140 (2004), 952–992.

    Article  MathSciNet  MATH  Google Scholar 

  102. S. J. Miller, Variation in the number of points on elliptic curves and applications to excess rank, C. R. Math. Rep. Acad. Sci. Canada 27 (2005), no. 4, 111–120.

    MathSciNet  MATH  Google Scholar 

  103. S. J. Miller (with an appendix by E. Dueñez), Investigations of zeros near the central point of elliptic curve L-functions, Experimental Mathematics 15 (2006), no. 3, 257–279.

    Google Scholar 

  104. S. J. Miller, Lower order terms in the 1-level density for families of holomorphic cuspidal newforms, Acta Arithmetica 137 (2009), 51–98.

    Article  MathSciNet  MATH  Google Scholar 

  105. S. J. Miller and R. Peckner, Low-lying zeros of number field L-functions, Journal of Number Theory 132 (2012), 2866–2891.

    Article  MathSciNet  MATH  Google Scholar 

  106. S. J. Miller and R. Takloo-Bighash, An Invitation to Modern Number Theory, Princeton University Press, Princeton, NJ, 2006.

    MATH  Google Scholar 

  107. H. Montgomery, The pair correlation of zeros of the zeta function, Analytic Number Theory, Proc. Sympos. Pure Math. 24, Amer. Math. Soc., Providence, 1973, 181–193.

    Google Scholar 

  108. H. Montgomery and P. Weinberger, Notes on small class numbers, Acta Arith. 24 (1973) 529–542.

    MathSciNet  MATH  Google Scholar 

  109. A. Odlyzko, On the distribution of spacings between zeros of the zeta function, Math. Comp. 48 (1987), no. 177, 273–308.

    Article  MathSciNet  MATH  Google Scholar 

  110. A. Odlyzko, The 10 22 -nd zero of the Riemann zeta function, Proc. Conference on Dynamical, Spectral and Arithmetic Zeta-Functions, M. van Frankenhuysen and M. L. Lapidus, eds., Amer. Math. Soc., Contemporary Math. series, 2001. http://www.research.att.com/~amo/doc/zeta.html.

  111. A. M. Odlyzko, New analytic algorithms in number theory, Proc. Intern. Congress Math. 1986, Amer. Math. Soc., 1987, pp. 466–475.

    Google Scholar 

  112. A. M. Odlyzko, The 10 20 -th zero of the Riemann zeta function and 70 million of its neighbors (1989), unpublished.

    Google Scholar 

  113. A. M. Odlyzko, The 10 20 -th zero of the Riemann zeta function and 175 million of its neighbors (1992), unpublished.

    Google Scholar 

  114. A. M. Odlyzko and A. Schönhage, Fast algorithms for multiple evalutions of the Riemann zeta function, Trans. Amer. Math. Soc. 309 (1988), 797–809.

    Article  MathSciNet  MATH  Google Scholar 

  115. J. Oesterlé, Nombre de classes des corps quadratiques imaginaires, Séminaire Nicolas Bourbaki, Vol. 1983/84, Astérisque No. 121–122 (1985), 309–323.

    Google Scholar 

  116. A. E. Özlük and C. Snyder, Small zeros of quadratic L-functions, Bull. Austral. Math. Soc. 47 (1993), no. 2, 307–319.

    Article  MathSciNet  MATH  Google Scholar 

  117. A. E. Özlük and C. Snyder, On the distribution of the nontrivial zeros of quadratic L-functions close to the real axis, Acta Arith. 91 (1999), no. 3, 209–228.

    MathSciNet  MATH  Google Scholar 

  118. M. Ostrofsky, G. Breit, and D. P. Johnson, The Excitation Function of Lithium Under Proton Bombardment, Phys. Rev. 49 (1936), 22–34.

    Article  Google Scholar 

  119. C. Porter (editor), Statistical Theories of Spectra: Fluctuations, Academic Press, New York, 1965.

    Google Scholar 

  120. C. Porter and N. Rosenzweig, Repulsion of energy levels in complex atomic spectra, Phys. Rev. 120 (1960) 1698–1714.

    Article  Google Scholar 

  121. L. J. Rainwater, Nuclear Energy Level Argument for a Spheroidal Nuclear Model, Phys. Rev. 79 (1950), 432–434.

    Article  MATH  Google Scholar 

  122. G. Ricotta and E. Royer, Statistics for low-lying zeros of symmetric power L-functions in the level aspect, Forum Math. 23 (2011), no. 5, 969–1028.

    Article  MathSciNet  MATH  Google Scholar 

  123. G. F. B. Riemann, Über die Anzahl der Primzahlen unter einer gegebenen Grösse, Monatsber. Königl. Preuss. Akad. Wiss. Berlin, Nov. 1859, 671–680 (see [39] for an English translation).

    Google Scholar 

  124. J. Rosen, J. S. Desjardins, J. Rainwater and W. Havens Jr., Slow neutron resonance spectroscopy I, Phys. Rev. 118 (1960) 687–697.

    Article  Google Scholar 

  125. E. Royer, Petits zéros de fonctions L de formes modulaires, Acta Arith. 99 (2001), 47–172.

    Article  MathSciNet  Google Scholar 

  126. M. Rubinstein, Low-lying zeros of L-functions and Random Matrix Theory, Duke Math. J. 109 (2001), no. 1, 147–181.

    Article  MathSciNet  MATH  Google Scholar 

  127. Z. Rudnick and P. Sarnak, Zeros of principal L-functions and Random Matrix Theory, Duke Math. J. 81 (1996), 269–322.

    Article  MathSciNet  MATH  Google Scholar 

  128. A. Selberg, An Elementary Proof of the Prime Number Theorem, Ann. Math. 50 (1949), 305–313.

    Article  MathSciNet  MATH  Google Scholar 

  129. J. P. Serre, A Course in Arithmetic, Springer-Verlag, New York, 1996.

    MATH  Google Scholar 

  130. S. W. Shin and N. Templier, Sato-Tate theorem for families and low-lying zeros of automorphic L-functions, Appendix A by Robert Kottwitz, and Appendix B by Raf Cluckers, Julia Gordon and Immanuel Halupczok. Invent. Math. 203 (2016), no. 1, 1–177.

    Article  MathSciNet  Google Scholar 

  131. C. L. Siegel, Über Riemanns Nachlass zur analytischen Zahlentheorie, Quellen und Studien zur Geschichte der Math. Astr. Phys., no. 2, 1932, pp. 45–80; reprinted in C.L. Siegel, Gesammelte Abhandlungen, Vol. 1, Springer, 1966.

    Google Scholar 

  132. C.L. Siegel, Über die Klassenzahl quadratischer Zahlkörper, Acta. Arith. 1 (1935) 83–86.

    MATH  Google Scholar 

  133. H. M. Stark, A complete determination of the complex quadratic fields of class-number one, Michigan Math. J. 14 (1969) 1–27.

    MathSciNet  MATH  Google Scholar 

  134. H. M. Stark, A transcendence theorem for class number problems, Ann. of Math. 2 (1971) 153–173.

    Article  MathSciNet  MATH  Google Scholar 

  135. J. Stopple, Notes on the Deuring-Heilbronn phenomenon, Notices Amer. Math. Soc. 53 (2006), no. 8, 864–875

    MathSciNet  MATH  Google Scholar 

  136. T. Tao, Topics in Random Matrix Theory, Graduate Studies in Mathematics, volume 132, American Mathematical Society, Providence, RI 2012.

    Google Scholar 

  137. T. Tao and V. Vu, From the Littlewood-Offord problem to the Circular Law: universality of the spectral distribution of random matrices, Bull. Amer. Math. Soc. 46 (2009), 377–396.

    Article  MathSciNet  MATH  Google Scholar 

  138. T. Tao and V. Vu, Random matrices: universality of local eigenvalue statistics up to the edge, Comm. Math. Phys. 298, (2010), no. 2, 549–572

    Article  MathSciNet  MATH  Google Scholar 

  139. A. M. Turing, Some calculations of the Riemann zeta-function, Proc. London Math. Soc., ser. 3 3 (1953), 99–117.

    Google Scholar 

  140. C. Wanger, Class number 5, 6, and 7, Math. Comp. 65 (1996), no. 214, 785–800.

    Article  MathSciNet  Google Scholar 

  141. M. Watkins, Class numbers of imaginary quadratic fields, Mathematics of Computation 73 (2004), 907–938.

    Article  MathSciNet  MATH  Google Scholar 

  142. W. Weibull, A statistical distribution function of wide applicability, J. Appl. Mech. Trans. ASME. 18 (1951) 293–297.

    MATH  Google Scholar 

  143. E. Wigner, On the statistical distribution of the widths and spacings of nuclear resonance levels, Proc. Cambridge Philo. Soc. 47 (1951), 790–798.

    Article  MATH  Google Scholar 

  144. E. Wigner, Characteristic vectors of bordered matrices with infinite dimensions, Ann. of Math. 2 (1955), no. 62, 548–564.

    Article  MathSciNet  MATH  Google Scholar 

  145. E. Wigner, Statistical Properties of real symmetric matrices. Pages 174–184 in Canadian Mathematical Congress Proceedings, University of Toronto Press, Toronto, 1957.

    Google Scholar 

  146. E. Wigner, Characteristic vectors of bordered matrices with infinite dimensions. II, Ann. of Math. Ser. 2 65 (1957), 203–207.

    Article  MathSciNet  MATH  Google Scholar 

  147. E. Wigner, Results and theory of resonance absorption, Gatlinburg Conference on Neutron Physics by Time-of-Flight, Oak Ridge National Lab. Report No. ORNL–2309 (1957), 59–70.

    Google Scholar 

  148. E. Wigner, On the distribution of the roots of certain symmetric matrices, Ann. of Math. Ser. 67 (1958), no. 2, 325–327.

    Article  MathSciNet  MATH  Google Scholar 

  149. Wikipedia Contributors, Biometrika, Wikipedia, The Free Encyclopedia, Date retrieved 29 March 2015 04:05 UTC, Page Version ID 609531691, http://en.wikipedia.org/w/index.php?title=Biometrika&oldid=609531691.

  150. J. Wishart, The generalized product moment distribution in samples from a normal multivariate population, Biometrika 20 A (1928), 32–52.

    Google Scholar 

  151. A. Yang, Low-lying zeros of Dedekind zeta functions attached to cubic number fields, preprint.

    Google Scholar 

  152. M. Young, Low-lying zeros of families of elliptic curves, J. Amer. Math. Soc. 19 (2006), no. 1, 205–250.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The third named author was partially supported by NSF grant DMS1265673. We thank our colleagues and collaborators over the years for many helpful discussions on these and related topics. One of us (Miller) was fortunate to be a graduate student at Princeton, and had numerous opportunities then to converse with John Nash on a variety of mathematical topics. It was always a joy sitting next to him at seminars. We are grateful for his kind invitation to contribute to this work, and his comments on an earlier draft.

Author information

Authors and Affiliations

  1. University of Chicago, Chicago, IL, 60637, USA

    Owen Barrett

  2. Yale University, New Haven, CT, 06520, USA

    Frank W. K. Firk

  3. Williams College, Williamstown, MA, 01267, USA

    Steven J. Miller

  4. Duke University, Durham, NC, 27708, USA

    Caroline Turnage-Butterbaugh

Authors
  1. Owen Barrett
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Frank W. K. Firk
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Steven J. Miller
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Caroline Turnage-Butterbaugh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Steven J. Miller .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Senior Research Mathematician, Princeton University, Princeton, New Jersey, USA

    John Forbes Nash, Jr.

  2. Department of Mathematics, Princeton University, Princeton, New Jersey, USA

    Michael Th. Rassias

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Barrett, O., Firk, F.W.K., Miller, S.J., Turnage-Butterbaugh, C. (2016). From Quantum Systems to L-Functions: Pair Correlation Statistics and Beyond. In: Nash, Jr., J., Rassias, M. (eds) Open Problems in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-32162-2_2

Download citation

Publish with us

Policies and ethics

Search

Navigation