Low-lying zeros of L-functions with orthogonal symmetry

Duke Mathematical Journal

Volumn 136, Issue 1, 2007, Pages 115-172

  Hughes, C P ^a,b   Miller, Steven J ^c  

^a UNIVERSITY OF MICHIGAN   (United States)

^b UNIVERSITY OF YORK   (United Kingdom)

^c BROWN UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 33846270739     PISSN: 00127094     EISSN: None     Source Type: Journal    
DOI: 10.1215/S0012-7094-07-13614-7     Document Type: Article

References (27)

1. 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), 33-104. MR 2149530
2. E. DUEÑEZ and S. J. MILLER, The low lying zeros of a GL(4) and a GL(6) family of L-functions, to appear in Compos. Math., preprint, arXiv:math.NT/0506462
3. E. FOUVRY and H. IWANIEC, Low-lying zeros of dihedral L-functions, Duke Math. J. 116 (2003), 189-217. MR 1953291
4. P. GAO, n-level density of the low-lying zeros of quadratic Dirichlet L-functions, Ph.D. dissertation, University of Michigan, Ann Arbor, 2005.
5. I. S. GRADSHTEYN and I. M. RYZHIK, Tables of Integrals, Series, and Products, 4th ed., Academic Press, New York, 1965. MR 0197789
6. A. GÜLOǦLU, Low-lying zeros of symmetric power L-functions, Int. Math. Res. Not. 2005, no. 9, 517-550. MR 2131448
7. R. K. GUY, Catwalks, sandsteps and Pascal pyramids, J. Integer Seq. 3 (2000), article 00.1.6. MR 1750743
8. D. A. HEJHAL, On the triple correlation of zeros of the zeta function, Internat. Math. Res. Notices 1994, no. 7, 293-302. MR 1283025
9. J. HOFFSTEIN and P. LOCKHART, Coefficients of Maass forms and the Siegel zero, with an appendix by D. Goldfeld, J. Hoffstein, and D. Lieman, Ann. of Math. (2) 140 (1994), 161-181. MR 1289494
10. C. P. HUGHES and Z. RUDNICK, Linear statistics of low-lying zeros of L-functions, Q. J. Math. 54 (2003), 309-333. MR 2013141
11. _, Mock-Gaussian behaviour for linear statistics of classical compact groups, J. Phys. A 36 (2003), 2919-2932. MR 1986399
12. 0(N) , Acta Arith. 56 (1990), 65-82. MR 1067982
13. _, Topics in Classical Automorphic Forms, Grad. Stud. Math. 17, Amer. Math. Soc., Providence, 1997. MR 1474964
14. 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. MR 1828743
15. N. M. KATZ and P. SARNAK, Random Matrices, Frobenius Eigenvalues, and Monodromy, Amer. Math. Soc. Colloq. Publ. 45, Amer. Math. Soc., Providence, 1999. MR 1659828
16. _, Zeros of zeta functions and symmetry. Bull. Amer. Math. Soc. (N.S.) 36 (1999), 1-26. MR 1640151
17. J. P. KEATING and N. C. SNAITH, Random matrices and L-functiuns, J. Phys. A 36 (2003), 2859-2881. MR 1986396
18. S. J. MILLER, One- and two-level densities for families of elliptic curves: Evidence for the underlying group symmetries, Compos. Math. 104 (2004), 952-992. MR 2059225
19. _, 1- and 2-leveI densities for families of elliptic curves: Evidence for the underlying group symmetries, Ph.D. dissertation, Princeton University, Princeton, N.J., 2002.
20. H. L. MONTGOMERY, "The pair correlation of zeros of the zeta function" in Analytic Number Theory (St. Louis, Mo., 1972), Amer. Math. Soc., Providence, 1973, 181-193. MR 0337821
21. E. ROYER, Petits zéros de fonctions L deformes modulaires, Acta Arith. 99 (2001), 147-172. MR 1847618
22. M. RUBINSTEIN, Low-lying zeros of L-functions and random matrix theory, Duke Math. J. 109 (2001), 147-181. MR 1844208
23. Z. RUDNICK and P. SARNAK, Zeros of principal L-functions and random matrix theory, Duke Math. J. 81 (1996), 269-322. MR 1395406
24. A. SOSHNIKOV, Central limit theorem for local linear statistics in classical compact groups and related combinatorial identities, Ann. Probab. 28 (2000), 1353-1370. MR 1797877
25. E. C. TITCHMARSH, The Theory of the Riemann Zeta-Function, 2nd ed., Oxford Univ. Press, New York, 1986. MR 0882550
26. G. N. WATSON, A Treatise on the Theory of Bessel Functions, reprint of the 2nd ed., Cambridge Math. Lib., Cambridge Univ. Press, 1995. MR 1349110
27. M. P. YOUNG, Low-lying zeros of families of elliptic curves, J. Amer. Math. Soc. 19 (2006), 205-250. MR 2169047