Orthogonal and symplectic 𝑛-level densities

Mason, A.; Snaith, N.

doi:10.1090/memo/1194

AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution

Orthogonal and symplectic $n$-level densities

About this Title

A. M. Mason, NDM Experimental Medicine, University of Oxford, John Radcliffe Hospital, Oxford, OX3 9DU, United Kingdom and N. C. Snaith, School of Mathematics, University of Bristol, Bristol, BS8 1TW, United Kingdom

Publication: Memoirs of the American Mathematical Society
Publication Year: 2018; Volume 251, Number 1194
ISBNs: 978-1-4704-2685-9 (print); 978-1-4704-4262-0 (online)
DOI: https://doi.org/10.1090/memo/1194
Published electronically: September 11, 2017
MSC: Primary 11M50, 15B52, 11M26, 11G05, 11M06, 15B10

View full volume PDF

Abstract

In this paper we apply to the zeros of families of $L$-functions with orthogonal or symplectic symmetry the method that Conrey and Snaith (Correlations of eigenvalues and Riemann zeros, 2008) used to calculate the $n$-correlation of the zeros of the Riemann zeta function. This method uses the Ratios Conjectures (Conrey, Farmer, and Zimbauer, 2008) for averages of ratios of zeta or $L$-functions. Katz and Sarnak (Zeroes of zeta functions and symmetry, 1999) conjecture that the zero statistics of families of $L$-functions have an underlying symmetry relating to one of the classical compact groups $U(N)$, $O(N)$ and $USp(2N)$. Here we complete the work already done with $U(N)$ (Conrey and Snaith, Correlations of eigenvalues and Riemann zeros, 2008) to show how new methods for calculating the $n$-level densities of eigenangles of random orthogonal or symplectic matrices can be used to create explicit conjectures for the $n$-level densities of zeros of $L$-functions with orthogonal or symplectic symmetry, including all the lower order terms. We show how the method used here results in formulae that are easily modified when the test function used has a restricted range of support, and this will facilitate comparison with rigorous number theoretic $n$-level density results.

References

M. Abramowitz and I. A. Stegun. Handbook of mathematical functions: with formulas, graphs, and mathematical tables, volume 55. Dover publications, 55, 1965.
Matthew W. Alderson and Michael O. Rubinstein, Conjectures and experiments concerning the moments of $L(1/2,\chi _d)$, Exp. Math. 21 (2012), no. 3, 307–328. MR 2988582, DOI 10.1080/10586458.2012.687238
George Arfken, Mathematical methods for physicists, Academic Press, New York-London, 1966. MR 0205512
E. W. Barnes, The theory of the $G$-function, Q. J. Math., 31:264–314, 1900.
E. B. Bogomolny and J. P. Keating, Random matrix theory and the Riemann zeros. I. Three- and four-point correlations, Nonlinearity 8 (1995), no. 6, 1115–1131. MR 1363402
E. B. Bogomolny and J. P. Keating. Gutzwiller’s trace formula and spectral statistics: beyond the diagonal approximation. Phys. Rev. Lett., 77(8):1472–1475, 1996.
E. B. Bogomolny and J. P. Keating, Random matrix theory and the Riemann zeros. II. $n$-point correlations, Nonlinearity 9 (1996), no. 4, 911–935. MR 1399479, DOI 10.1088/0951-7715/9/4/006
E. Bogomolny and J. P. Keating, A method for calculating spectral statistics based on random-matrix universality with an application to the three-point correlations of the Riemann zeros, J. Phys. A 46 (2013), no. 30, 305203, 17. MR 3080328, DOI 10.1088/1751-8113/46/30/305203
Enrico Bombieri, The Riemann hypothesis, The millennium prize problems, Clay Math. Inst., Cambridge, MA, 2006, pp. 107–124. MR 2238277
Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor, On the modularity of elliptic curves over $\mathbf Q$: wild 3-adic exercises, J. Amer. Math. Soc. 14 (2001), no. 4, 843–939. MR 1839918, DOI 10.1090/S0894-0347-01-00370-8
Edouard Brézin and Shinobu Hikami, Characteristic polynomials of random matrices, Comm. Math. Phys. 214 (2000), no. 1, 111–135. MR 1794268, DOI 10.1007/s002200000256
Daniel Bump and Alex Gamburd, On the averages of characteristic polynomials from classical groups, Comm. Math. Phys. 265 (2006), no. 1, 227–274. MR 2217304, DOI 10.1007/s00220-006-1503-1
Jasbir S. Chahal and Brian Osserman, The Riemann hypothesis for elliptic curves, Amer. Math. Monthly 115 (2008), no. 5, 431–442. MR 2408488, DOI 10.1080/00029890.2008.11920545
Joseph A. Cima, Alec L. Matheson, and William T. Ross, The Cauchy transform, Mathematical Surveys and Monographs, vol. 125, American Mathematical Society, Providence, RI, 2006. MR 2215991
J. Brian Conrey, $L$-functions and random matrices, Mathematics unlimited—2001 and beyond, Springer, Berlin, 2001, pp. 331–352. MR 1852163
Brian Conrey, Notes on eigenvalue distributions for the classical compact groups, Recent perspectives in random matrix theory and number theory, London Math. Soc. Lecture Note Ser., vol. 322, Cambridge Univ. Press, Cambridge, 2005, pp. 111–145. MR 2166460, DOI 10.1017/CBO9780511550492.005
J. B. Conrey and D. W. Farmer, Mean values of $L$-functions and symmetry, Internat. Math. Res. Notices 17 (2000), 883–908. MR 1784410, DOI 10.1155/S1073792800000465
J. B. Conrey, D. W. Farmer, J. P. Keating, M. O. Rubinstein, and N. C. Snaith, Autocorrelation of random matrix polynomials, Comm. Math. Phys. 237 (2003), no. 3, 365–395. MR 1993332, DOI 10.1007/s00220-003-0852-2
J. B. Conrey, D. W. Farmer, J. P. Keating, M. O. Rubinstein, and N. C. Snaith, Integral moments of $L$-functions, Proc. London Math. Soc. (3) 91 (2005), no. 1, 33–104. MR 2149530, DOI 10.1112/S0024611504015175
J. B. Conrey, D. W. Farmer, and M. R. Zirnbauer. Autocorrelation of ratios of characteristic polynomials and of $L$-functions. preprint.
Brian Conrey, David W. Farmer, and Martin R. Zirnbauer, Autocorrelation of ratios of $L$-functions, Commun. Number Theory Phys. 2 (2008), no. 3, 593–636. MR 2482944, DOI 10.4310/CNTP.2008.v2.n3.a4
J. B. Conrey, P. J. Forrester, and N. C. Snaith, Averages of ratios of characteristic polynomials for the compact classical groups, Int. Math. Res. Not. 7 (2005), 397–431. MR 2130839, DOI 10.1155/IMRN.2005.397
J. B. Conrey and A. Ghosh, On the Selberg class of Dirichlet series: small degrees, Duke Math. J. 72 (1993), no. 3, 673–693. MR 1253620, DOI 10.1215/S0012-7094-93-07225-0
J. B. Conrey and A. Ghosh, Mean values of the Riemann zeta-function. III, Proceedings of the Amalfi Conference on Analytic Number Theory (Maiori, 1989) Univ. Salerno, Salerno, 1992, pp. 35–59. MR 1220455
J. B. Conrey and S. M. Gonek, High moments of the Riemann zeta-function, Duke Math. J. 107 (2001), no. 3, 577–604. MR 1828303, DOI 10.1215/S0012-7094-01-10737-0
J. B. Conrey and N. C. Snaith, Applications of the $L$-functions ratios conjectures, Proc. Lond. Math. Soc. (3) 94 (2007), no. 3, 594–646. MR 2325314, DOI 10.1112/plms/pdl021
John Brian Conrey and Nina Claire Snaith, Correlations of eigenvalues and Riemann zeros, Commun. Number Theory Phys. 2 (2008), no. 3, 477–536. MR 2482941, DOI 10.4310/CNTP.2008.v2.n3.a1
J. B. Conrey and N. C. Snaith, In support of $n$-correlation, Comm. Math. Phys. 330 (2014), no. 2, 639–653. MR 3223483, DOI 10.1007/s00220-014-1969-1
J. Brian Conrey and Nina C. Snaith, Triple correlation of the Riemann zeros, J. Théor. Nombres Bordeaux 20 (2008), no. 1, 61–106 (English, with English and French summaries). MR 2434158
Adrian Diaconu, Dorian Goldfeld, and Jeffrey Hoffstein, Multiple Dirichlet series and moments of zeta and $L$-functions, Compositio Math. 139 (2003), no. 3, 297–360. MR 2041614, DOI 10.1023/B:COMP.0000018137.38458.68
Eduardo Dueñez and Steven J. Miller, The low lying zeros of a $\rm GL(4)$ and a $\rm GL(6)$ family of $L$-functions, Compos. Math. 142 (2006), no. 6, 1403–1425. MR 2278752, DOI 10.1112/S0010437X0600220X
Freeman J. Dyson, Statistical theory of the energy levels of complex systems. I, J. Mathematical Phys. 3 (1962), 140–156. MR 143556, DOI 10.1063/1.1703773
Martin Eichler, Quaternäre quadratische Formen und die Riemannsche Vermutung für die Kongruenzzetafunktion, Arch. Math. 5 (1954), 355–366 (German). MR 63406, DOI 10.1007/BF01898377
Alexei Entin, Edva Roditty-Gershon, and Zeév Rudnick, Low-lying zeros of quadratic Dirichlet L-functions, hyper-elliptic curves and random matrix theory, Geom. Funct. Anal. 23 (2013), no. 4, 1230–1261. MR 3077912, DOI 10.1007/s00039-013-0241-8
David W. Farmer, Modeling families of $L$-functions, Ranks of elliptic curves and random matrix theory, London Math. Soc. Lecture Note Ser., vol. 341, Cambridge Univ. Press, Cambridge, 2007, pp. 53–69. MR 2322337, DOI 10.1017/CBO9780511735158.003
Daniel Fiorilli and Steven J. Miller, Surpassing the ratios conjecture in the 1-level density of Dirichlet $L$-functions, Algebra Number Theory 9 (2015), no. 1, 13–52. MR 3317760, DOI 10.2140/ant.2015.9.13
A. Florea. Improving the error term in the mean value of $L(\tfrac {1}{2},\chi )$ in the hyper elliptic ensemble. preprint, arXiv:1505.03094.
E. Fouvry and H. Iwaniec, Low-lying zeros of dihedral $L$-functions, Duke Math. J. 116 (2003), no. 2, 189–217. MR 1953291, DOI 10.1215/S0012-7094-03-11621-X
Peng Gao, n-level density of the low-lying zeros of quadratic Dirichlet L-functions, ProQuest LLC, Ann Arbor, MI, 2005. Thesis (Ph.D.)–University of Michigan. MR 2708011
Peng Gao, $n$-level density of the low-lying zeros of quadratic Dirichlet $L$-functions, Int. Math. Res. Not. IMRN 6 (2014), 1699–1728. MR 3180607, DOI 10.1093/imrn/rns261
John Goes, Steven Jackson, Steven J. Miller, David Montague, Kesinee Ninsuwan, Ryan Peckner, and Thuy Pham, A unitary test of the ratios conjecture, J. Number Theory 130 (2010), no. 10, 2238–2258. MR 2660890, DOI 10.1016/j.jnt.2010.02.020
Ahmet Muhtar Güloğlu, Low-lying zeroes of symmetric power $L$-functions, Int. Math. Res. Not. 9 (2005), 517–550. MR 2131448, DOI 10.1155/IMRN.2005.517
G. H. Hardy and J. E. Littlewood, Contributions to the theory of the riemann zeta-function and the theory of the distribution of primes, Acta Math. 41 (1916), no. 1, 119–196. MR 1555148, DOI 10.1007/BF02422942
D. R. Heath-Brown, Prime number theory and the Riemann zeta-function, Recent perspectives in random matrix theory and number theory, London Math. Soc. Lecture Note Ser., vol. 322, Cambridge Univ. Press, Cambridge, 2005, pp. 1–30. MR 2166457, DOI 10.1017/CBO9780511550492.002
Dennis A. Hejhal, On the triple correlation of zeros of the zeta function, Internat. Math. Res. Notices 7 (1994), 293ff., approx. 10 pp.}, issn=1073-7928, review= MR 1283025, doi=10.1155/S1073792894000334,
C. P. Hughes and Steven J. Miller, Low-lying zeros of $L$-functions with orthogonal symmetry, Duke Math. J. 136 (2007), no. 1, 115–172. MR 2271297, DOI 10.1215/S0012-7094-07-13614-7
C. P. Hughes and Z. Rudnick, Linear statistics of low-lying zeros of $L$-functions, Q. J. Math. 54 (2003), no. 3, 309–333. MR 2013141, DOI 10.1093/qjmath/54.3.309
D. K. Huynh, J. P. Keating, and N. C. Snaith, Lower order terms for the one-level density of elliptic curve $L$-functions, J. Number Theory 129 (2009), no. 12, 2883–2902. MR 2560841, DOI 10.1016/j.jnt.2008.12.008
Duc Khiem Huynh, Steven J. Miller, and Ralph Morrison, An elliptic curve test of the $L$-functions ratios conjecture, J. Number Theory 131 (2011), no. 6, 1117–1147. MR 2786594, DOI 10.1016/j.jnt.2010.12.004
Adrian Diaconu, Dorian Goldfeld, and Jeffrey Hoffstein, Multiple Dirichlet series and moments of zeta and $L$-functions, Compositio Math. 139 (2003), no. 3, 297–360. MR 2041614, DOI 10.1023/B:COMP.0000018137.38458.68
A. E. Ingham, Mean-Value Theorems in the Theory of the Riemann Zeta-Function, Proc. London Math. Soc. (2) 27 (1927), no. 4, 273–300. MR 1575391, DOI 10.1112/plms/s2-27.1.273
Henryk Iwaniec and Emmanuel Kowalski, Analytic number theory, American Mathematical Society Colloquium Publications, vol. 53, American Mathematical Society, Providence, RI, 2004. MR 2061214
Henryk Iwaniec, Wenzhi Luo, and Peter Sarnak, Low lying zeros of families of $L$-functions, Inst. Hautes Études Sci. Publ. Math. 91 (2000), 55–131 (2001). MR 1828743
Alan Jeffrey and Hui-Hui Dai, Handbook of mathematical formulas and integrals, 4th ed., Elsevier/Academic Press, Amsterdam, 2008. With 1 CD-ROM (Windows and Macintosh). MR 2400410
M. Jutila, On the mean value of $L({1\over 2},\,\chi )$ for real characters, Analysis 1 (1981), no. 2, 149–161. MR 632705, DOI 10.1524/anly.1981.1.2.149
Nicholas M. Katz and Peter Sarnak, Random matrices, Frobenius eigenvalues, and monodromy, American Mathematical Society Colloquium Publications, vol. 45, American Mathematical Society, Providence, RI, 1999. MR 1659828
Nicholas M. Katz and Peter Sarnak, Zeroes of zeta functions and symmetry, Bull. Amer. Math. Soc. (N.S.) 36 (1999), no. 1, 1–26. MR 1640151, DOI 10.1090/S0273-0979-99-00766-1
J. P. Keating and N. C. Snaith, Random matrices and $L$-functions, J. Phys. A 36 (2003), no. 12, 2859–2881. Random matrix theory. MR 1986396, DOI 10.1088/0305-4470/36/12/301
J. P. Keating and N. C. Snaith, Random matrix theory and $L$-functions at $s=1/2$, Comm. Math. Phys. 214 (2000), no. 1, 91–110. MR 1794267, DOI 10.1007/s002200000262
J. P. Keating and N. C. Snaith, Random matrix theory and $\zeta (1/2+it)$, Comm. Math. Phys. 214 (2000), no. 1, 57–89. MR 1794265, DOI 10.1007/s002200000261
Neal Koblitz, Introduction to elliptic curves and modular forms, 2nd ed., Graduate Texts in Mathematics, vol. 97, Springer-Verlag, New York, 1993. MR 1216136
Steven G. Krantz, Function theory of several complex variables, John Wiley & Sons, Inc., New York, 1982. Pure and Applied Mathematics; A Wiley-Interscience Publication. MR 635928
E. G. Ladopoulos Universal Fracture Mechanics by the Generalized Sokhotski-Plemelj Formulas for Finite-Part Singular Integrals Universal Journal of Fracture Mechanics, 1:17-26, 2013
Jake Levinson and Steven J. Miller, The $n$-level densities of low-lying zeros of quadratic Dirichlet $L$-functions, Acta Arith. 161 (2013), no. 2, 145–182. MR 3141916, DOI 10.4064/aa161-2-3
A. M. Mason. Correlations of zeros of families of $L$-functions with orthogonal or symplectic symmetry. PhD thesis, University of Bristol, 2013.
Madan Lal Mehta, Random matrices, 3rd ed., Pure and Applied Mathematics (Amsterdam), vol. 142, Elsevier/Academic Press, Amsterdam, 2004. MR 2129906
Steven J. Miller, One- and two-level densities for rational families of elliptic curves: evidence for the underlying group symmetries, Compos. Math. 140 (2004), no. 4, 952–992. MR 2059225, DOI 10.1112/S0010437X04000582
Steven J. Miller, A symplectic test of the $L$-functions ratios conjecture, Int. Math. Res. Not. IMRN 3 (2008), Art. ID rnm146, 36. MR 2416994, DOI 10.1093/imrn/rnm146
Steven J. Miller, An orthogonal test of the $L$-functions ratios conjecture, Proc. Lond. Math. Soc. (3) 99 (2009), no. 2, 484–520. MR 2533673, DOI 10.1112/plms/pdp009
Steven J. Miller and David Montague, An orthogonal test of the $L$-functions ratios conjecture, II, Acta Arith. 146 (2011), no. 1, 53–90. MR 2741191, DOI 10.4064/aa146-1-5
Steven J. Miller and Ryan Peckner, Low-lying zeros of number field $L$-functions, J. Number Theory 132 (2012), no. 12, 2866–2891. MR 2965197, DOI 10.1016/j.jnt.2012.05.034
H. L. Montgomery, The pair correlation of zeros of the zeta function, Analytic number theory (Proc. Sympos. Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo., 1972) Amer. Math. Soc., Providence, R.I., 1973, pp. 181–193. MR 0337821
A. M. Odlyzko, The $10^{20}$th zero of the Riemann zeta function and 70 million of its neighbors, preprint, 1989, http://www.dtc.umn.edu/˜odlyzko/unpublished/index.html.
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. MR 1735673, DOI 10.4064/aa-91-3-209-228
J. Plemelj, Ein Ergänzungssatz zur Cauchyschen Integraldarstellung analytischer Funktionen, Randwerte betreffend, Monatsh. Math. Phys. 19 (1908), no. 1, 205–210 (German). MR 1547763, DOI 10.1007/BF01736696
Guillaume Ricotta and Emmanuel Royer, Lower order terms for the one-level densities of symmetric power $L$-functions in the level aspect, Acta Arith. 141 (2010), no. 2, 153–170. MR 2579842, DOI 10.4064/aa141-2-2
Emmanuel Royer, Petits zéros de fonctions $L$ de formes modulaires, Acta Arith. 99 (2001), no. 2, 147–172 (French). MR 1847618, DOI 10.4064/aa99-2-3
Michael Oded Rubinstein, Evidence for a spectral interpretation of the zeros of L-functions, ProQuest LLC, Ann Arbor, MI, 1998. Thesis (Ph.D.)–Princeton University. MR 2697494
Michael Rubinstein, Low-lying zeros of $L$-functions and random matrix theory, Duke Math. J. 109 (2001), no. 1, 147–181. MR 1844208, DOI 10.1215/S0012-7094-01-10916-2
Zeév Rudnick and Peter Sarnak, Zeros of principal $L$-functions and random matrix theory, Duke Math. J. 81 (1996), no. 2, 269–322. A celebration of John F. Nash, Jr. MR 1395406, DOI 10.1215/S0012-7094-96-08115-6
Goro Shimura, Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, No. 11. Iwanami Shoten, Publishers, Tokyo; Princeton University Press, Princeton, N.J., 1971. Kanô Memorial Lectures, No. 1. MR 0314766
Joseph H. Silverman, The arithmetic of elliptic curves, 2nd ed., Graduate Texts in Mathematics, vol. 106, Springer, Dordrecht, 2009. MR 2514094
N. C. Snaith, Riemann zeros and random matrix theory, Milan J. Math. 78 (2010), no. 1, 135–152. MR 2684776, DOI 10.1007/s00032-010-0114-7
Y. V. Sokhotski On definite integrals and functions utilized for expansions into series. St Petersburg, 1873
K. Soundararajan, Nonvanishing of quadratic Dirichlet $L$-functions at $s=\frac 12$, Ann. of Math. (2) 152 (2000), no. 2, 447–488. MR 1804529, DOI 10.2307/2661390
Richard Taylor and Andrew Wiles, Ring-theoretic properties of certain Hecke algebras, Ann. of Math. (2) 141 (1995), no. 3, 553–572. MR 1333036, DOI 10.2307/2118560
Lawrence C. Washington, Elliptic curves, 2nd ed., Discrete Mathematics and its Applications (Boca Raton), Chapman & Hall/CRC, Boca Raton, FL, 2008. Number theory and cryptography. MR 2404461
Andrew Wiles, Modular elliptic curves and Fermat’s last theorem, Ann. of Math. (2) 141 (1995), no. 3, 443–551. MR 1333035, DOI 10.2307/2118559
Matthew P. Young, Low-lying zeros of families of elliptic curves, J. Amer. Math. Soc. 19 (2006), no. 1, 205–250. MR 2169047, DOI 10.1090/S0894-0347-05-00503-5
Matthew P. Young, The third moment of quadratic Dirichlet L-functions, Selecta Math. (N.S.) 19 (2013), no. 2, 509–543. MR 3090236, DOI 10.1007/s00029-012-0104-4
Qiao Zhang, On the cubic moment of quadratic Dirichlet $L$-functions, Math. Res. Lett. 12 (2005), no. 2-3, 413–424. MR 2150894, DOI 10.4310/MRL.2005.v12.n3.a11

AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution

Orthogonal and symplectic $n$-level densities

About this Title

Table of Contents

Abstract

References [Enhancements On Off] (What's this?)

memo_has_moved_text();Orthogonal and symplectic $n$-level densities

About this Title

Table of Contents

Abstract

References [Enhancements On Off] (What's this?)

Orthogonal and symplectic $n$-level densities