{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T12:49:12Z","timestamp":1773233352500,"version":"3.50.1"},"reference-count":133,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2009,9,20]],"date-time":"2009-09-20T00:00:00Z","timestamp":1253404800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this article, we discuss the remarkable connection between two very different fields, number theory and nuclear physics. We describe the essential aspects of these fields, the quantities studied, and how insights in one have been fruitfully applied in the other. The exciting branch of modern mathematics \u2013 random matrix theory \u2013 provides the connection between the two fields. We assume no detailed knowledge of number theory, nuclear physics, or random matrix theory; all that is required is some familiarity with linear algebra and probability theory, as well as some results from complex analysis. Our goal is to provide the inquisitive reader with a sound overview of the subjects, placing them in their historical context in a way that is not traditionally given in the popular and technical surveys.<\/jats:p>","DOI":"10.3390\/sym1010064","type":"journal-article","created":{"date-parts":[[2009,9,21]],"date-time":"2009-09-21T08:23:52Z","timestamp":1253521432000},"page":"64-105","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":33,"title":["Nuclei, Primes and the Random Matrix Connection"],"prefix":"10.3390","volume":"1","author":[{"given":"Frank W. K.","family":"Firk","sequence":"first","affiliation":[{"name":"The Henry Koerner Center for Emeritus Faculty, Yale University, New Haven, CT 06520, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Steven J.","family":"Miller","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, Williams College, Williamstown, MA 01267, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2009,9,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"32","DOI":"10.1093\/biomet\/20A.1-2.32","article-title":"The generalized product moment distribution in samples from a normal multivariate population","volume":"20A","author":"Wishart","year":"1928","journal-title":"Biometrika"},{"key":"ref_2","unstructured":"Wigner, E. Results and theory of resonance absorption. Gatlinburg Conference on Neutron Physics by Time-of-Flight, Oak Ridge, National Lab, Report No. ORNL\u20132309, 1957."},{"key":"ref_3","unstructured":"Montgomery, H. (1973). Analytic Number Theory, Proceedings of the Symposium on Pure Mathematicsm, Amer. Math. Soc."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"296","DOI":"10.1511\/2003.26.296","article-title":"The spectrum of Riemannium","volume":"91","author":"Hayes","year":"2003","journal-title":"Am. Sci."},{"key":"ref_5","unstructured":"Rockmore, D. (2005). Stalking the Riemann Hypothesis: The Quest to Find the Hidden Law of Prime Numbers, Pantheon."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Conrey, J.B. (2001). Mathematics unlimited \u2014 2001 and Beyond, Springer-Verlag. L-Functions and random matrices.","DOI":"10.1007\/978-3-642-56478-9_14"},{"key":"ref_7","first-page":"341","article-title":"The Riemann hypothesis","volume":"50","author":"Conrey","year":"2003","journal-title":"Notices of the AMS"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"R1","DOI":"10.1088\/0305-4470\/36\/12\/201","article-title":"Developments in Random Matrix Theory","volume":"36","author":"Forrester","year":"2003","journal-title":"J. Phys. A"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1090\/S0273-0979-99-00766-1","article-title":"Zeros of zeta functions and symmetries","volume":"36","author":"Katz","year":"1999","journal-title":"Bull. AMS"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"2859","DOI":"10.1088\/0305-4470\/36\/12\/301","article-title":"Random matrices and L-functions","volume":"36","author":"Keating","year":"2003","journal-title":"J. Phys. A: Math. Gen."},{"key":"ref_11","unstructured":"Mehta, M.L. (2004). Random Matrices, Elsevier. [3rd ed.]."},{"key":"ref_12","unstructured":"Hardy, G.H., and Wright, E. (1995). An Introduction to the Theory of Numbers, Oxford Science Publications, Clarendon Press. [5th ed.]."},{"key":"ref_13","unstructured":"Sloane, N.J.A. The On-Line Encyclopedia of Integer Sequences. available online at http:\/\/www.research.att.com\/\u223cnjas\/sequences\/A000945."},{"key":"ref_14","unstructured":"Erd\u00f6s, P. (1949). D\u00e9monstration \u00e9l\u00e9mentaire du th\u00e9ore#x2018;me sur la distribution des nombres premiers, Centre Mathe#x2018;matique. Scriptum 1."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"305","DOI":"10.2307\/1969455","article-title":"An Elementary Proof of the Prime Number Theorem","volume":"50","author":"Selberg","year":"1949","journal-title":"Ann. Math."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Miller, S.J., and Takloo-Bighash, R. (2006). An Invitation to Modern Number Theory, Princeton University Press.","DOI":"10.1515\/9780691215976"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Davenport, H. (1980). Graduate Texts in Mathematics Series: Multiplicative Number Theory, Springer-Verlag. [2nd ed.]. revised by Montgomery, H.","DOI":"10.1007\/978-1-4757-5927-3"},{"key":"ref_18","unstructured":"Riemann, G.F.B. (1859). \u00dcber die Anzahl der Primzahlen unter einer gegebenen Gr\u00f6sse. Monatsber. K\u00f6nigl. Preuss. Akad. Wiss. Berlin, 671\u2013680. See [19] for an English translation."},{"key":"ref_19","unstructured":"Edwards, H.M. (1974). Riemann\u2019s Zeta Function, Dover Publications."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Lang, S. (1999). Graduate Texts in Mathematics: Complex Analysis, Springer-Verlag.","DOI":"10.1007\/978-1-4757-3083-8"},{"key":"ref_21","unstructured":"Stein, E., and Shakarchi, R. (2003). Complex Analysis, Princeton University Press."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"199","DOI":"10.24033\/bsmf.545","article-title":"Sur la distribution des z\u00e9ros de la fonction \u03b6(s) et ses cons\u00e9quences arithm\u00e9tiques","volume":"24","author":"Hadamard","year":"1896","journal-title":"Bull. Soc. math. France"},{"key":"ref_23","first-page":"183","article-title":"Recherches analytiques la th\u00e9orie des nombres premiers","volume":"20","year":"1896","journal-title":"Ann. Soc. scient. Brux."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Chudnovsky, D., Chudnovsky, G., and Nathanson, M. (2004). Number Theory, New York Seminar 2003, Springer-Verlag.","DOI":"10.1007\/978-1-4419-9060-0"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"385","DOI":"10.1103\/RevModPhys.53.385","article-title":"Random-matrix physics: spectrum and strength fluctuations","volume":"53","author":"Brody","year":"1981","journal-title":"Rev. Mod. Phys."},{"key":"ref_26","unstructured":"Derrien, H., Leal, L., and Larson, N. (2004). Status of new evaluation of the neutron resonance parameters of 238U ORNL. PHYSOR 2004, Chicago, USA, 25\u201329 April, 2004, Amer. Nucl. Soc.. Available on CD-ROM."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"140","DOI":"10.1063\/1.1703773","article-title":"Statistical theory of the energy levels of complex systems: I, II, III","volume":"3","author":"Dyson","year":"1962","journal-title":"J. Mathematical Phys."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1199","DOI":"10.1063\/1.1703863","article-title":"The threefold way. Algebraic structure of symmetry groups and ensembles in quantum mechanics","volume":"3","author":"Dyson","year":"1962","journal-title":"J. Mathematical Phys."},{"key":"ref_29","unstructured":"Firk, F., Lynn, J.E., and Moxon, M. (-, January Aug). Parameters of neutron resonances in U238 below 1.8 keV. Proceedings of the International Conference on Nuclear Structure, Kingston, Toronto."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"309","DOI":"10.1016\/0369-643X(58)90081-1","article-title":"High resolution neutron time- of-flight experiments using the Harwell 15 MeV linear electron accelerator","volume":"3","author":"Firk","year":"1958","journal-title":"Nucl. Instr."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"2313","DOI":"10.1103\/PhysRevLett.54.2313","article-title":"Bounds on time-reversal non-invariance in the nuclear Hamiltonian","volume":"54","author":"French","year":"1985","journal-title":"Phys. Rev. Lett."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"447","DOI":"10.1016\/0029-5582(61)90176-6","article-title":"Sur la loi limite de l\u2019espacement des valeurs propres d\u2019une matrice al\u00e9atoire","volume":"25","author":"Gaudin","year":"1961","journal-title":"Nucl. Phys."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"471","DOI":"10.1103\/PhysRev.109.471","article-title":"Spacings of nuclear energy levels","volume":"109","author":"Harvey","year":"1958","journal-title":"Phys. Rev."},{"key":"ref_34","first-page":"1086","article-title":"Fluctuation properties of nuclear energy levels: do theory and experiment agree?","volume":"48","author":"Haq","year":"1982","journal-title":"Phys. Rev."},{"key":"ref_35","unstructured":"Hughes, D. (1957). Neutron Cross Sections, Pergamon Press."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1016\/0029-5582(60)90413-2","article-title":"On the statistical properties of level spacings in nuclear spectra","volume":"18","author":"Mehta","year":"1960","journal-title":"Nucl. Phys."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"420","DOI":"10.1016\/0029-5582(60)90414-4","article-title":"On the density of the eigenvalues of a random matrix","volume":"18","author":"Mehta","year":"1960","journal-title":"Nucl. Phys."},{"key":"ref_38","unstructured":"Porter, C. (1965). Statistical Theories of Spectra: Fluctuations, Academic Press."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"790","DOI":"10.1017\/S0305004100027237","article-title":"On the statistical distribution of the widths and spacings of nuclear resonance levels","volume":"47","author":"Wigner","year":"1951","journal-title":"Proc. Cambridge Phil. Soc."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"548","DOI":"10.2307\/1970079","article-title":"Characteristic vectors of bordered matrices with infinite dimensions","volume":"2","author":"Wigner","year":"1955","journal-title":"Ann. of Math."},{"key":"ref_41","unstructured":"Wigner, E. (1957). Canadian Mathematical Congress Proceedings, University of Toronto Press."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"203","DOI":"10.2307\/1969956","article-title":"Characteristic vectors of bordered matrices with infinite dimensions. II","volume":"65","author":"Wigner","year":"1957","journal-title":"Ann. of Math. Ser. 2"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"325","DOI":"10.2307\/1970008","article-title":"On the distribution of the roots of certain symmetric matrices","volume":"67","author":"Wigner","year":"1958","journal-title":"Ann. of Math. Ser. 2"},{"key":"ref_44","unstructured":"Casella, G., and Berger, R. (2002). Statistical Inference, Duxbury Advanced Series. [2nd ed.]."},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Shohat, J.A., and Tamarkin, J.D. (1943). The Problem of Moments, AMS.","DOI":"10.1090\/surv\/001"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"82","DOI":"10.1006\/aima.1998.1728","article-title":"The classical moment problem as a self-adjoint finite difference operator","volume":"137","author":"Simon","year":"1998","journal-title":"Adv. Math."},{"key":"ref_47","unstructured":"Baik, J., Borodin, A., Deift, P., and Suidan, T. (2005). A Model for the Bus System in Cuernevaca (Mexico). Math. Phys., 1\u20139. Online available at http:\/\/arxiv.org\/abs\/math\/0510414."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"L229","DOI":"10.1088\/0305-4470\/33\/26\/102","article-title":"The statistical properties of the city transport in Cuernavaca (Mexico) and Random matrix ensembles","volume":"33","author":"Krbalek","year":"2000","journal-title":"J. Phys. A: Math. Gen"},{"key":"ref_49","unstructured":"Whittaker, E. (1944). A Treatise on the Analytical Dynamics of Particles and Rigid Bodies: With an Introduction to the Problem of Three Bodies, Cambridge University Press."},{"key":"ref_50","unstructured":"Reif, F. (1965). Fundamentals of Statistical and Thermal Physics, McGraw-Hill."},{"key":"ref_51","first-page":"1348","article-title":"\u201cWhat is a random matrix?\u201d","volume":"52","author":"Diaconis","year":"2005","journal-title":"Notices Amer. Math. Soc."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1090\/S0273-0979-03-00975-3","article-title":"Patterns of Eigenvalues: the 70th Josiah Willard Gibbs Lecture","volume":"40","author":"Diaconis","year":"2003","journal-title":"Bull. Amer. Math. Soc."},{"key":"ref_53","first-page":"539","article-title":"Neutron Time-of-Flight Spectrometers","volume":"162","author":"Bromley","year":"1979","journal-title":"Detectors in Nuclear Science"},{"key":"ref_54","unstructured":"Harvey, J.A. (1970). Experimental Neutron Resonance Spectroscopy, Academic Press."},{"key":"ref_55","unstructured":"Lynn, J.E. (1968). The Theory of Neutron Resonance Reactions, The Clarendon Press."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"687","DOI":"10.1103\/PhysRev.118.687","article-title":"Slow neutron resonance spectroscopy I","volume":"118","author":"Rosen","year":"1960","journal-title":"Phys. Rev."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"2214","DOI":"10.1103\/PhysRev.120.2214","article-title":"Slow neutron resonance spectroscopy II","volume":"120","author":"Desjardins","year":"1960","journal-title":"Phys. Rev."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"985","DOI":"10.1103\/PhysRev.134.B985","article-title":"Neutron resonance spectroscopy III","volume":"134","author":"Garg","year":"1964","journal-title":"Th 232 and U238. Phys. Rev."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"1132","DOI":"10.1016\/S0031-8914(56)90086-6","article-title":"Repulsion of nuclear levels","volume":"22","author":"Gurevich","year":"1956","journal-title":"Physica"},{"key":"ref_60","unstructured":"Landau, L., and Smorodinski, Ya. (1955). Lektsii po teorii atomnogo yadra. Gos Izd. tex - teoreyicheskoi Lit. Moscow, 92\u201394."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"293","DOI":"10.1115\/1.4010337","article-title":"A statistical distribution function of wide applicability","volume":"18","author":"Weibull","year":"1951","journal-title":"J. Appl. Mech. Trans."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"40","DOI":"10.1080\/09332480.2007.10722831","article-title":"A derivation of the Pythagorean Won-Loss Formula in baseball","volume":"20","author":"Miller","year":"2007","journal-title":"Chance Magazine"},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"1698","DOI":"10.1103\/PhysRev.120.1698","article-title":"Repulsion of energy levels in complex atomic spectra","volume":"120","author":"Porter","year":"1960","journal-title":"Phys. Rev."},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1103\/PhysRevLett.52.1","article-title":"Characterization of chaotic quantum spectra and universality of level fluctuation laws","volume":"52","author":"Bohigas","year":"1984","journal-title":"Phys. Rev. Lett."},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"895","DOI":"10.1103\/RevModPhys.72.895","article-title":"The Statistical Theory of Quantum Dots","volume":"72","author":"Alhassid","year":"2000","journal-title":"Rev. Mod. Phys."},{"key":"ref_66","first-page":"1012","article-title":"Sur les Z\u00e9ros de la Fonction \u03b6(s) de Riemann","volume":"158","author":"Hardy","year":"1914","journal-title":"C. R. Acad. Sci. Paris"},{"key":"ref_67","first-page":"1","article-title":"On the zeros of Riemann\u2019s zeta-function","volume":"10","author":"Selberg","year":"1942","journal-title":"Skr. Norske Vid. Akad. Oslo I."},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"383","DOI":"10.1016\/0001-8708(74)90074-7","article-title":"More than one-third of the zeros of Riemann\u2019s zeta function are on \u03c3 = 1\/2","volume":"13","author":"Levinson","year":"1974","journal-title":"Adv. In Math."},{"key":"ref_69","first-page":"1","article-title":"More than two fifths of the zeros of the Riemann zeta function are on the critical line","volume":"399","author":"Conrey","year":"1989","journal-title":"J. Reine angew. Math."},{"key":"ref_70","first-page":"247","article-title":"The Gauss Class-Number Problems","volume":"7","author":"Stark","year":"2007","journal-title":"Clay Math. Proc."},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"907","DOI":"10.1090\/S0025-5718-03-01517-5","article-title":"Class numbers of imaginary quadratic fields","volume":"73","author":"Watkins","year":"2004","journal-title":"Math. Comp."},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"259","DOI":"10.4064\/aa103-3-5","article-title":"Spacing of Zeros of Hecke L-Functions and the Class Number Problem","volume":"103","author":"Conrey","year":"2002","journal-title":"Acta Arith"},{"key":"ref_73","first-page":"624","article-title":"The class number of quadratic fields and the conjectures of Birch and Swinnerton-Dyer","volume":"4","author":"Goldfeld","year":"1976","journal-title":"Ann. Scuola Norm. Sup. Pisa Cl. Sci. 3"},{"key":"ref_74","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1007\/BF01388809","article-title":"Heegner points and derivatives of L-series","volume":"84","author":"Gross","year":"1986","journal-title":"Invent. Math."},{"key":"ref_75","doi-asserted-by":"crossref","first-page":"273","DOI":"10.1090\/S0025-5718-1987-0866115-0","article-title":"On the distribution of spacings between zeros of the zeta function","volume":"48","author":"Odlyzko","year":"1987","journal-title":"Math. Comp."},{"key":"ref_76","unstructured":"van Frankenhuysen, M., and Lapidus, M.L. The 1022-nd zero of the Riemann zeta function. Proceedings Conference on Dynamical, Spectral and Arithmetic Zeta-Function, Online available at http:\/\/www.research.att.com\/~amo\/doc\/zeta.html."},{"key":"ref_77","unstructured":"Erd\u00f6s, L., Ramirez, J.A., Schlein, B., and Yau, H.-T. Bulk Universality for Wigner Matrices, Preprint: http:\/\/arxiv.org\/abs\/0905.4176."},{"key":"ref_78","unstructured":"Erd\u00f6s, L., Schlein, B., and Yau, H.-T. Wegner estimate and level repulsion for Wigner random matrices, Preprint: http:\/\/arxiv.org\/abs\/0905.4176."},{"key":"ref_79","doi-asserted-by":"crossref","first-page":"377","DOI":"10.1090\/S0273-0979-09-01252-X","article-title":"From the Littlewood-Offord problem to the Circular Law: universality of the spectral distribution of random matrices","volume":"46","author":"Tao","year":"2009","journal-title":"Bull. Amer. Math. Soc."},{"key":"ref_80","unstructured":"Tao, T., and Vu, V. Random matrices: universality of local eigenvalue statistics up to the edge, Comm. Math. Phys., Preprint: http:\/\/arxiv.org\/PS cache\/arxiv\/pdf\/0908\/0908.1982v1.pdf."},{"key":"ref_81","doi-asserted-by":"crossref","unstructured":"Grimmett, G.R., and Stirzaker, D.R. (2001). Probability and Random Processes, Oxford University Press. [3rd ed.].","DOI":"10.1093\/oso\/9780198572237.001.0001"},{"key":"ref_82","unstructured":"Feller, W. (1971). An Introduction to Probability Theory and Its Applications, John Wiley & Sons. [2nd ed.]."},{"key":"ref_83","doi-asserted-by":"crossref","first-page":"537","DOI":"10.1007\/s10959-005-3518-5","article-title":"Eigenvalue spacing distribution for the ensemble of real symmetric Toeplitz matrices","volume":"18","author":"Hammond","year":"2005","journal-title":"J. Theoret. Probab."},{"key":"ref_84","doi-asserted-by":"crossref","first-page":"637","DOI":"10.1007\/s10959-007-0078-x","article-title":"Eigenvalue spacing distribution for the ensemble of real symmetric palindromic Toeplitz matrices","volume":"20","author":"Massey","year":"2007","journal-title":"J. Theor. Probab."},{"key":"ref_85","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1215\/S0012-7094-01-10916-2","article-title":"Low-lying zeros of L-functions and random matrix theory","volume":"109","author":"Rubinstein","year":"2001","journal-title":"Duke Math. J."},{"key":"ref_86","doi-asserted-by":"crossref","first-page":"269","DOI":"10.1215\/S0012-7094-96-08115-6","article-title":"Zeros of principal L-functions and random matrix theory","volume":"81","author":"Rudnick","year":"1996","journal-title":"Duke Math. J."},{"key":"ref_87","unstructured":"Gao, P. (2005). N-level density of the low-lying zeros of quadratic Dirichlet L-functions. [Ph.D. Dissertation, University of Michigan]."},{"key":"ref_88","doi-asserted-by":"crossref","unstructured":"Stanley, R.P. (1999). Enumerative Combinatorics, Cambridge University Press. [ed. 2].","DOI":"10.1017\/CBO9780511609589"},{"key":"ref_89","unstructured":"Lehman, R. (2001). First order spacings of random matrix eigenvalues. [Junior\u2019s Thesis, Princeton University]."},{"key":"ref_90","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1016\/0024-3795(81)90150-6","article-title":"The expected eigenvalue distribution of a large regular graph","volume":"40","author":"McKay","year":"1981","journal-title":"Linear Algebra Appl."},{"key":"ref_91","unstructured":"Jakobson, D., Miller, S.D., Rivin, I., and Rudnick, Z. (1999). Emerging Applications of Number Theory, Vol. 109, Springer."},{"key":"ref_92","doi-asserted-by":"crossref","first-page":"231","DOI":"10.1080\/10586458.2008.10129029","article-title":"The distribution of the second largest eigenvalue in families of random regular graphs","volume":"17","author":"Miller","year":"2008","journal-title":"Exp. Mat."},{"key":"ref_93","doi-asserted-by":"crossref","first-page":"151","DOI":"10.1007\/BF02100489","article-title":"Level-spacing distributions and the Airy kernel","volume":"159","author":"Tracy","year":"1994","journal-title":"Commun. Math. Phys."},{"key":"ref_94","doi-asserted-by":"crossref","unstructured":"Tracy, C.A., and Widom, H. (1996). On Orthogonal and Sympletic Matrix Ensembles. Commun. Math. Phys., 727\u2013754.","DOI":"10.1007\/BF02099545"},{"key":"ref_95","unstructured":"Tatsien, L.I. (2002, January Aug). Distribution functions for largest eigenvalues and their applications. Proceedings of the ICM Vol. I, Beijing, China."},{"key":"ref_96","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1214\/009117905000000495","article-title":"Spectral measure of large random Hankel, Markov and Toeplitz matrices","volume":"34","author":"Bryc","year":"2006","journal-title":"Ann. Probab."},{"key":"ref_97","doi-asserted-by":"crossref","unstructured":"Hejhal, D. (1994). On the triple correlation of zeros of the zeta function. Int. Math. Res. Notices, 294\u2013302.","DOI":"10.1155\/S1073792894000334"},{"key":"ref_98","doi-asserted-by":"crossref","unstructured":"Katz, N., and Sarnak, P. (1999). Random Matrices, Frobenius Eigenvalues and Monodromy, AMS Colloquium Publications.","DOI":"10.1090\/coll\/045"},{"key":"ref_99","doi-asserted-by":"crossref","unstructured":"Iwaniec, H., and Kowalski, E. (2004). Analytic Number Theory, AMS Colloquium Publications.","DOI":"10.1090\/coll\/053"},{"key":"ref_100","doi-asserted-by":"crossref","first-page":"952","DOI":"10.1112\/S0010437X04000582","article-title":"1- and 2-level densities for families of elliptic curves: evidence for the underlying group symmetries","volume":"140","author":"Miller","year":"2004","journal-title":"Compositio Mathematica"},{"key":"ref_101","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1215\/S0012-7094-07-13614-7","article-title":"Low-lying zeros of L-functions with orthogonal symmetry","volume":"136","author":"Hughes","year":"2007","journal-title":"Duke Math. J."},{"key":"ref_102","doi-asserted-by":"crossref","first-page":"1403","DOI":"10.1112\/S0010437X0600220X","article-title":"The low lying zeros of a GL(4) and a GL(6) family of L-functions","volume":"142","author":"Miller","year":"2006","journal-title":"Compositio Mathematica"},{"key":"ref_103","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1215\/S0012-7094-03-11621-X","article-title":"Low-lying zeros of dihedral L-functions","volume":"116","author":"Fouvry","year":"2003","journal-title":"Duke Math. J."},{"key":"ref_104","doi-asserted-by":"crossref","unstructured":"G\u00fclo\u011flu, A. (2005). Low Lying Zeros of Symmetric Power L-Functions. Int. Math. Res. Not., 517\u2013550.","DOI":"10.1155\/IMRN.2005.517"},{"key":"ref_105","doi-asserted-by":"crossref","first-page":"309","DOI":"10.1093\/qmath\/hag021","article-title":"Linear statistics of low-lying zeros of L-functions","volume":"54","author":"Hughes","year":"2003","journal-title":"Q. J. Math."},{"key":"ref_106","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1007\/BF02698741","article-title":"Low lying zeros of families of L-functions","volume":"91","author":"Iwaniec","year":"2000","journal-title":"Inst. Hautes \u00c9tudes Sci. Publ. Math."},{"key":"ref_107","doi-asserted-by":"crossref","first-page":"51","DOI":"10.4064\/aa137-1-3","article-title":"Lower order terms in the 1-level density for families of holomorphic cuspidal newforms","volume":"137","author":"Miller","year":"2009","journal-title":"Acta Arith."},{"key":"ref_108","doi-asserted-by":"crossref","first-page":"47","DOI":"10.4064\/aa99-2-3","article-title":"Petits z\u00e9ros de fonctions L de formes modulaires","volume":"99","author":"Royer","year":"2001","journal-title":"Acta Arith."},{"key":"ref_109","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1090\/S0894-0347-05-00503-5","article-title":"Low-lying zeros of families of elliptic curves","volume":"19","author":"Young","year":"2006","journal-title":"J. Amer. Math. Soc."},{"key":"ref_110","doi-asserted-by":"crossref","first-page":"577","DOI":"10.1215\/S0012-7094-01-10737-0","article-title":"High moments of the Riemann zeta-function","volume":"107","author":"Conrey","year":"2001","journal-title":"Duke Math. Jour."},{"key":"ref_111","doi-asserted-by":"crossref","first-page":"775","DOI":"10.1155\/S1073792898000476","article-title":"A conjecture for the sixth power moment of the Riemann zeta-function","volume":"15","author":"Conrey","year":"1998","journal-title":"Int. Math. Res. Not."},{"key":"ref_112","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1112\/S0024611504015175","article-title":"Integral moments of L-functions","volume":"91","author":"Conrey","year":"2005","journal-title":"Proc. London Math. Soc."},{"key":"ref_113","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1007\/s002200000261","article-title":"Random matrix theory and \u03b6(1\/2+it)","volume":"214","author":"Keating","year":"2000","journal-title":"Commun. Math. Phys."},{"key":"ref_114","doi-asserted-by":"crossref","first-page":"91","DOI":"10.1007\/s002200000262","article-title":"Random matrix theory and L-functions at s = 1\/2","volume":"214","author":"Keating","year":"2000","journal-title":"Commun. Math. Phys."},{"key":"ref_115","doi-asserted-by":"crossref","unstructured":"Borwein, P., Choi, S., and Rooney, B. (2008). The Riemann Hypothesis, Springer. CMS Books in Mathematics.","DOI":"10.1007\/978-0-387-72126-2"},{"key":"ref_116","unstructured":"Conrey, J.B., Farmer, D.W., and Zirnbauer, M.R. Autocorrelation of ratios of L-functions. Preprint: http:\/\/arxiv.org\/abs\/0711.0718."},{"key":"ref_117","unstructured":"Conrey, J.B., Farmer, D.W., and Zirnbauer, M.R. Howe pairs, supersymmetry, and ratios of random characteristic polynomials for the classical compact groups. Preprint: http:\/\/arxiv.org\/abs\/math-ph\/0511024."},{"key":"ref_118","doi-asserted-by":"crossref","first-page":"594","DOI":"10.1112\/plms\/pdl021","article-title":"Applications of the L-functions Ratios Conjecture","volume":"94","author":"Conrey","year":"2007","journal-title":"Proc. London Math. Soc."},{"key":"ref_119","unstructured":"Due\u00f1ez, E., and Miller, S.J. (2009). Proc. London Math. Soc."},{"key":"ref_120","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1080\/10586458.1996.10504583","article-title":"\u00c9tude exp\u00e9rimentale du rang de familles de courbes elliptiques sur \u211a","volume":"5","author":"Fermigier","year":"1996","journal-title":"Exper. Math."},{"key":"ref_121","first-page":"93","article-title":"Sur la loi de probabilite de l\u2019ecart maximum","volume":"6","author":"Frechet","year":"1927","journal-title":"Ann. de la Soc. Polonaise de Mathematique"},{"key":"ref_122","unstructured":"Goes, J., Jackson, S., Miller, S.J., Montague, D., Ninsuwan, K., Peckner, R., and Pham, T. A unitary test of the Ratios Conjecture, Preprint."},{"key":"ref_123","doi-asserted-by":"crossref","first-page":"507","DOI":"10.1215\/S0012-7094-07-13634-2","article-title":"A hybrid Euler-Hadamard product for the Riemann zeta function","volume":"136","author":"Gonek","year":"2007","journal-title":"Duke Math. J."},{"key":"ref_124","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1080\/10586458.2006.10128967","article-title":"Investigations of zeros near the central point of elliptic curve L-functions","volume":"15","author":"Miller","year":"2006","journal-title":"Experim. Math."},{"key":"ref_125","unstructured":"Miller, S.J. (2008). A symplectic test of the L-Functions Ratios Conjecture. Int. Math. Res. Notices."},{"key":"ref_126","unstructured":"Miller, S.J. (2009). Proceedings of the London Mathematical Society, Part2."},{"key":"ref_127","unstructured":"Miller, S.J., and Montague, D. An Orthogonal Test of the L-functions Ratios Conjecture, II, Preprint."},{"key":"ref_128","unstructured":"Montgomery, H., and Soundararajan, K. (2002). Paul Erd\u00f6s and His Mathematics, I (Vol. 11), Bolyai Society Mathematical Studies."},{"key":"ref_129","unstructured":"Ricotta, G., and Royer, E. Statistics for low-lying zeros of symmetric power L-functions in the level aspect. Preprint: http:\/\/arxiv.org\/abs\/math\/0703760."},{"key":"ref_130","doi-asserted-by":"crossref","unstructured":"Sarnak, P. (1990). Cambridge Trusts in Mathemetics: Some applications of modular forms, Cambridge University Press.","DOI":"10.1017\/CBO9780511895593"},{"key":"ref_131","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1080\/10586458.2009.10128899","article-title":"The quadratic character experiment","volume":"18","author":"Stopple","year":"2009","journal-title":"Exp. Math."},{"key":"ref_132","unstructured":"Wormald, N.C. (1999). London Mathematical Society Lecture Note Series: Surveys in combinatorics (Canterbury), Cambridge University Press."},{"key":"ref_133","doi-asserted-by":"crossref","first-page":"587","DOI":"10.1155\/IMRN.2005.587","article-title":"Lower-order terms of the 1-level density of families of elliptic curves","volume":"10","author":"Young","year":"2005","journal-title":"Int. Math. Res. Not."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/1\/1\/64\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T22:11:12Z","timestamp":1760220672000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/1\/1\/64"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,9,20]]},"references-count":133,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2009,9]]}},"alternative-id":["sym1010064"],"URL":"https:\/\/doi.org\/10.3390\/sym1010064","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2009,9,20]]}}}