## RANDOM POLYNOMIALS, RANDOM MATRICES, AND L-FUNCTIONS, II

Citations: | 2 - 0 self |

### BibTeX

@MISC{Farmer_randompolynomials,,

author = {David W Farmer and Francesco Mezzadri and Nina and C Snaith},

title = {RANDOM POLYNOMIALS, RANDOM MATRICES, AND L-FUNCTIONS, II},

year = {}

}

### OpenURL

### Abstract

Abstract. We show that the Circular Orthogonal Ensemble of random matrices arises naturally from a family of random polynomials. This sheds light on the appearance of random matrix statistics in the zeros of the Riemann zeta-function. 1.

### Citations

638 |
Random Matrices
- Mehta
- 1991
(Show Context)
Citation Context ...re the randomness is explicitly encoded in the zeros. For example, consider the Weyl integration formula for the classical compact groups [24, 12] or the β = 1, 2, 4 ensembles of random matrix theory =-=[17]-=-. On the other hand, usually in the study of random polynomials (for a review of the subject see, e.g., Farahmand [6]) the randomness is explicitly encoded in the coefficients of the polynomials. Date... |

282 |
The Classical Groups
- Weyl
- 1946
(Show Context)
Citation Context ...al of a random matrix can be viewed as a random polynomial where the randomness is explicitly encoded in the zeros. For example, consider the Weyl integration formula for the classical compact groups =-=[24, 12]-=- or the β = 1, 2, 4 ensembles of random matrix theory [17]. On the other hand, usually in the study of random polynomials (for a review of the subject see, e.g., Farahmand [6]) the randomness is expli... |

148 |
Random Matrices, Frobenius Eigenvalues, and Monodromy
- Katz, Sarnak
- 1999
(Show Context)
Citation Context ... of the Riemann zetafunction and other L-functions [18, 19, 9, 22]. In addition, eigenvalues of other compact classical matrix groups give a good model of the zeros of various families of L-functions =-=[13, 12, 21, 11, 7]-=-. Furthermore, the characteristic polynomials of the matrices provide a good model of the L-functions themselves [16, 3, 15, 4, 10]. The L-functions studied in number theory are Dirichlet series havin... |

111 |
Zeros of principal L-functions and random matrix theory
- Rudnick, Sarnak
- 1996
(Show Context)
Citation Context ...tics of eigenvalues of unitary matrices, chosen uniformly with respect to Haar measure on U(N), are observed to closely match the statistics of zeros of the Riemann zetafunction and other L-functions =-=[18, 19, 9, 22]-=-. In addition, eigenvalues of other compact classical matrix groups give a good model of the zeros of various families of L-functions [13, 12, 21, 11, 7]. Furthermore, the characteristic polynomials o... |

103 | Zeroes of zeta functions and symmetry
- Katz, Sarnak
- 1999
(Show Context)
Citation Context ... of the Riemann zetafunction and other L-functions [18, 19, 9, 22]. In addition, eigenvalues of other compact classical matrix groups give a good model of the zeros of various families of L-functions =-=[13, 12, 21, 11, 7]-=-. Furthermore, the characteristic polynomials of the matrices provide a good model of the L-functions themselves [16, 3, 15, 4, 10]. The L-functions studied in number theory are Dirichlet series havin... |

53 |
Random matrix theory and L-functions at s
- Keating, Snaith
(Show Context)
Citation Context ...s give a good model of the zeros of various families of L-functions [13, 12, 21, 11, 7]. Furthermore, the characteristic polynomials of the matrices provide a good model of the L-functions themselves =-=[16, 3, 15, 4, 10]-=-. The L-functions studied in number theory are Dirichlet series having a functional equation and an Euler product. In this paper we are concerned with a wider class of Dirichlet series which have a fu... |

41 | Mean values of L-functions and symmetry
- Conrey, Farmer
- 2000
(Show Context)
Citation Context ...s give a good model of the zeros of various families of L-functions [13, 12, 21, 11, 7]. Furthermore, the characteristic polynomials of the matrices provide a good model of the L-functions themselves =-=[16, 3, 15, 4, 10]-=-. The L-functions studied in number theory are Dirichlet series having a functional equation and an Euler product. In this paper we are concerned with a wider class of Dirichlet series which have a fu... |

40 | The Riemann zeros and eigenvalue asymptotics
- Berry, Keating
- 1999
(Show Context)
Citation Context ...supports our previous observation. The appearance of CUE statistics for arithmetic L-functions has been compared to the appearance of CUE statistics in a chaotic system without time-reversal symmetry =-=[14, 1]-=-. Indeed, the appearance of the CUE statistics for zeros of L-functions has been heuristically explained by the analogy between the periodic orbit sum for the density of states of a classically chaoti... |

36 |
Random matrix theory and ζ(1/2+it
- Keating, Snaith
- 2000
(Show Context)
Citation Context ...s give a good model of the zeros of various families of L-functions [13, 12, 21, 11, 7]. Furthermore, the characteristic polynomials of the matrices provide a good model of the L-functions themselves =-=[16, 3, 15, 4, 10]-=-. The L-functions studied in number theory are Dirichlet series having a functional equation and an Euler product. In this paper we are concerned with a wider class of Dirichlet series which have a fu... |

35 |
On the triple correlation of zeros of the zeta function
- Hejhal
- 1994
(Show Context)
Citation Context ...tics of eigenvalues of unitary matrices, chosen uniformly with respect to Haar measure on U(N), are observed to closely match the statistics of zeros of the Riemann zetafunction and other L-functions =-=[18, 19, 9, 22]-=-. In addition, eigenvalues of other compact classical matrix groups give a good model of the zeros of various families of L-functions [13, 12, 21, 11, 7]. Furthermore, the characteristic polynomials o... |

34 | Random matrix theory and the derivative of the Riemann zeta function
- Hughes, Keating, et al.
(Show Context)
Citation Context |

33 | Evidence for a spectral interpretation of the zeros of L-functions
- Rubinstein
- 1998
(Show Context)
Citation Context ... of the Riemann zetafunction and other L-functions [18, 19, 9, 22]. In addition, eigenvalues of other compact classical matrix groups give a good model of the zeros of various families of L-functions =-=[13, 12, 21, 11, 7]-=-. Furthermore, the characteristic polynomials of the matrices provide a good model of the L-functions themselves [16, 3, 15, 4, 10]. The L-functions studied in number theory are Dirichlet series havin... |

32 |
Spectral Methods of Automorphic Forms
- Iwaniec
- 2002
(Show Context)
Citation Context ...enstein series on SL(2, Z): (A6) E(z; s) := 1∑′ y 2 s |mz + n| 2s. m,n Specifically, we have LQ(s) = 2(ay) −sE(z; s). The Eisenstein series E(z; s) is a well-understood object from number theory. See =-=[12]-=- for details. However, in that theory one usually fixes s and considers E(z; s) as a function of z. Indeed, as a function of z it has many fascinating properties: it is an eigenfunction of the hyperbo... |

30 | Quantum chaotic dynamics and random polynomials
- Bogomolny, Bohigas, et al.
- 1996
(Show Context)
Citation Context ... polynomials are equivalent to self-reciprocal polynomials, formula (1.4) also gives the number of the zeros of (1.2) that lie on the unit circle. In this context it was rederived by Bogomolny et al. =-=[2]-=- who also computed the two-point correlation function of such zeros. 1.2. L-functions. An L-function is a Dirichlet series, ∞� (1.5) L(s) = with an = Oɛ(n ɛ ) for every ɛ > 0, which has an analytic co... |

27 |
The 10 20th zero of the Riemann zeta function and 70 million of its neighbors
- Odlyzko
- 1989
(Show Context)
Citation Context ...tics of eigenvalues of unitary matrices, chosen uniformly with respect to Haar measure on U(N), are observed to closely match the statistics of zeros of the Riemann zetafunction and other L-functions =-=[18, 19, 9, 22]-=-. In addition, eigenvalues of other compact classical matrix groups give a good model of the zeros of various families of L-functions [13, 12, 21, 11, 7]. Furthermore, the characteristic polynomials o... |

16 |
The pair correlation of the zeta function
- Montgomery
- 1973
(Show Context)
Citation Context |

10 |
The Riemann Zeta function and quantum chaology
- Keating
- 1993
(Show Context)
Citation Context ...supports our previous observation. The appearance of CUE statistics for arithmetic L-functions has been compared to the appearance of CUE statistics in a chaotic system without time-reversal symmetry =-=[14, 1]-=-. Indeed, the appearance of the CUE statistics for zeros of L-functions has been heuristically explained by the analogy between the periodic orbit sum for the density of states of a classically chaoti... |

7 |
The number of real zeros of a random trigonometric polynomial
- Dunnage
- 1966
(Show Context)
Citation Context ...(1.1) to emphasize the comparison with characteristic polynomials of random matrices. When the coefficients cn and dn of the polynomial (1.3) are independent standard normal random variables, Dunnage =-=[5]-=- discovered that the expected number of real zeros is given by N (1.4) √ + O 3 � N 11/13 (log N) 3/13� . Since real trigonometric polynomials are equivalent to self-reciprocal polynomials, formula (1.... |

7 |
Topics in Random Polynomials
- Farahmand
- 1998
(Show Context)
Citation Context ...ical compact groups [24, 12] or the β = 1, 2, 4 ensembles of random matrix theory [17]. On the other hand, usually in the study of random polynomials (for a review of the subject see, e.g., Farahmand =-=[6]-=-) the randomness is explicitly encoded in the coefficients of the polynomials. Date: September 23, 2005. Research supported by the American Institute of Mathematics and the Focused Research Group gran... |

6 |
Low lying zeros of families of L-functions, Inst. Hautes tudes
- Iwaniec, Luo, et al.
(Show Context)
Citation Context |

4 |
Zeros of Epstein zeta functions and supercomputers
- Hejhal
- 1986
(Show Context)
Citation Context ... a functional equation and an Euler product. In this paper we are concerned with a wider class of Dirichlet series which have a functional equation but do not have an Euler product (see, for example, =-=[9, 10]-=-). It has been suggested that such functions can be modelled by random self-reciprocal polynomials [9]. In Appendix A we discuss the example of Epstein zeta functions. It is also possible to create Di... |

4 |
The classical groups”. Princeton University press 1946 Pseudogroups
- Weyl
- 1989
(Show Context)
Citation Context ...al of a random matrix can be viewed as a random polynomial where the randomness is explicitly encoded in the zeros. For example, consider the Weyl integration formula for the classical compact groups =-=[28, 15]-=- or the β = 1, 2, 4 ensembles of random matrix theory [20]. On the Date: February 7, 2008. Research supported by the American Institute of Mathematics and the Focused Research Group grant (0244660) fr... |

2 |
Maass forms and their L-functions, preprint
- Farmer, Lemurell
- 2005
(Show Context)
Citation Context |

2 | Conjugate reciprocal polynomials with all roots on the unit circle, preprint
- Petersen, Sinclair
- 2005
(Show Context)
Citation Context ...cients an which is uniform on a bounded disk containing C and zero outside. The following theorem gives the joint probability density function for the roots of such polynomials. Peterson and Sinclair =-=[20]-=- have found some interesting geometric properties of the coefficients of these polynomials. k=1s4 DAVID W FARMER, FRANCESCO MEZZADRI, AND NINA C SNAITH In what follows we denote by ∆(x1, . . . , xN) t... |

1 |
Multiplicative Distance Functions
- Sinclair
- 2005
(Show Context)
Citation Context ...e have, where ε is a quantity with modulus one that may vary at each occurrence, (3.11) JC = ε � � N−1 1 2 2 � � � � � � � ∂a1 ∂ρ1 . ∂aN ∂ρ1 ∂a1 ∂t1 . ∂aN ∂t1 · · · . .. · · · Following the method in =-=[23]-=-, for a given m, ∂a1 ∂ρM . ∂aN ∂ρM ∂a1 ∂tM . ∂aN ∂tM (3.12) an = (−1) n � βm e βm ′ n−2,m + (βm + 1 )e βm ′ n−1,m + e ′ � n,m , with the convention that (3.13a) (3.13b) (3.13c) e0 = e0,m = e ′ 0,m = 1... |

1 |
Epstein zeta functions and random polynomials, in preparation
- Farmer, Koutsoliotas
(Show Context)
Citation Context ... a functional equation and an Euler product. In this paper we are concerned with a wider class of Dirichlet series which have a functional equation but do not have an Euler product (see, for example, =-=[9, 10]-=-). It has been suggested that such functions can be modelled by random self-reciprocal polynomials [9]. In Appendix A we discuss the example of Epstein zeta functions. It is also possible to create Di... |

1 |
Joint densities of secular coefficients for unitary matrices
- Sommers, Haake, et al.
- 1998
(Show Context)
Citation Context ...oint probability density function for odd N is the same as that for eigenvalues of a randomly chosen matrix in the Circular Orthogonal Ensemble COE(N). Some aspects of this result can be derived from =-=[27]-=- but they approach the subject from a different perspective. j<k j<kRANDOM POLYNOMIALS AND L-FUNCTIONS 5 The theorem suggests that if a Dirichlet series with functional equation is chosen at random, ... |