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29
Random Matrix Theory and ζ(1/2 + it)
, 2000
"... We study the characteristic polynomials Z(U,#)of matrices U in the Circular Unitary Ensemble (CUE) of Random Matrix Theory. Exact expressions for any matrix size N are derived for the moments of and Z/Z # , and from these we obtain the asymptotics of the value distributions and cumulants of the re ..."
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Cited by 85 (15 self)
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We study the characteristic polynomials Z(U,#)of matrices U in the Circular Unitary Ensemble (CUE) of Random Matrix Theory. Exact expressions for any matrix size N are derived for the moments of and Z/Z # , and from these we obtain the asymptotics of the value distributions and cumulants of the real and imaginary parts of log Z as N ##. In the
The Riemann Zeros and Eigenvalue Asymptotics
 SIAM Rev
, 1999
"... Comparison between formulae for the counting functions of the heights t n of the Riemann zeros and of semiclassical quantum eigenvalues En suggests that the t n are eigenvalues of an (unknown) hermitean operator H, obtained by quantizing a classical dynamical system with hamiltonian H cl . Many feat ..."
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Cited by 40 (5 self)
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Comparison between formulae for the counting functions of the heights t n of the Riemann zeros and of semiclassical quantum eigenvalues En suggests that the t n are eigenvalues of an (unknown) hermitean operator H, obtained by quantizing a classical dynamical system with hamiltonian H cl . Many features of H cl are provided by the analogy; for example, the "Riemann dynamics" should be chaotic and have periodic orbits whose periods are multiples of logarithms of prime numbers. Statistics of the t n have a similar structure to those of the semiclassical En ; in particular, they display randommatrix universality at short range, and nonuniversal behaviour over longer ranges. Very refined features of the statistics of the t n can be computed accurately from formulae with quantum analogues. The RiemannSiegel formula for the zeta function is described in detail. Its interpretation as a relation between long and short periodic orbits gives further insights into the quantum spectral fluctuations. We speculate that the Riemann dynamics is related to the trajectories generated by the classical hamiltonian H cl = XP. Key words. spectral asymptotics, number theory AMS subject classifications. 11M26, 11M06, 35P20, 35Q40, 41A60, 81Q10, 81Q50 PII. S0036144598347497 1.
Random matrices and Lfunctions
 J. PHYS A MATH GEN
, 2003
"... In recent years there has been a growing interest in connections between the statistical properties of number theoretical Lfunctions and random matrix theory. We review the history of these connections, some of the major achievements and a number of applications. ..."
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Cited by 20 (7 self)
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In recent years there has been a growing interest in connections between the statistical properties of number theoretical Lfunctions and random matrix theory. We review the history of these connections, some of the major achievements and a number of applications.
APPLICATIONS OF THE LFUNCTIONS RATIOS CONJECTURES
 PROCEEDINGS OF THE LONDON MATHEMATICAL SOCIETY
, 2006
"... ..."
Determinantal processes with number variance saturation
 Comm. Math. Phys
, 2004
"... Abstract. Consider Dyson’s Hermitian Brownian motion model after a finite time S, where the process is started at N equidistant points on the real line. These N points after time S form a determinantal process and has a limit as N → ∞. This limting determinantal proceess has the interesting feature ..."
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Cited by 10 (0 self)
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Abstract. Consider Dyson’s Hermitian Brownian motion model after a finite time S, where the process is started at N equidistant points on the real line. These N points after time S form a determinantal process and has a limit as N → ∞. This limting determinantal proceess has the interesting feature that it shows number variance saturation. The variance of the number of particles in an interval converges to a limiting value as the length of the interval goes to infinity. Number variance saturation is also seen for example in the zeros of the Riemann ζfunction, [20], [2]. The process can also be constructed using nonintersecting paths and we consider several variants of this construction. One construction leads to a model which shows a transition from a nonuniversal behaviour with number variance saturation to a universal sinekernel behaviour as we go up the line. 1.
Spectral Statistics in the Quantized Cardioid Billiard
, 1994
"... : The spectral statistics in the strongly chaotic cardioid billiard are studied. The analysis is based on the first 11000 quantal energy levels for odd and even symmetry respectively. It is found that the levelspacing distribution is in good agreement with the GOE distribution of randommatrix the ..."
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Cited by 9 (7 self)
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: The spectral statistics in the strongly chaotic cardioid billiard are studied. The analysis is based on the first 11000 quantal energy levels for odd and even symmetry respectively. It is found that the levelspacing distribution is in good agreement with the GOE distribution of randommatrix theory. In case of the number variance and rigidity we observe agreement with the randommatrix model for shortrange correlations only, whereas for longrange correlations both statistics saturate in agreement with semiclassical expectations. Furthermore the conjecture that for classically chaotic systems the normalized mode fluctuations have a universal Gaussian distribution with unit variance is tested and found to be in very good agreement for both symmetry classes. By means of the Gutzwiller trace formula the trace of the cosinemodulated heat kernel is studied. Since the billiard boundary is focusing there are conjugate points giving rise to zeros at the locations of the periodic orbits in...
A symplectic test of the Lfunctions ratios conjecture
 Int. Math. Res. Notices, 2008, article ID rnm
"... ABSTRACT. Recently Conrey, Farmer and Zirnbauer [CFZ1, CFZ2] conjectured formulas for the averages over a family of ratios of products of shifted Lfunctions. Their Lfunctions Ratios Conjecture predicts both the main and lower order terms for many problems, ranging from nlevel correlations and den ..."
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Cited by 8 (3 self)
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ABSTRACT. Recently Conrey, Farmer and Zirnbauer [CFZ1, CFZ2] conjectured formulas for the averages over a family of ratios of products of shifted Lfunctions. Their Lfunctions Ratios Conjecture predicts both the main and lower order terms for many problems, ranging from nlevel correlations and densities to mollifiers and moments to vanishing at the central point. There are now many results showing agreement between the main terms of number theory and random matrix theory; however, there are very few families where the lower order terms are known. These terms often depend on subtle arithmetic properties of the family, and provide a way to break the universality of behavior. The Lfunctions Ratios Conjecture provides a powerful and tractable way to predict these terms. We test a specific case here, that of the 1level density for the symplectic family of quadratic Dirichlet characters arising from even fundamental discriminants d ≤ X. For test functions supported in (−1/3, 1/3) we calculate all the lower order terms up to size O(X −1/2+ǫ) and observe perfect agreement with the conjecture (for test functions supported in (−1, 1) we show agreement up to errors of size O(X −ǫ) for any ǫ). Thus for this family and suitably restricted test functions, we completely verify the Ratios Conjecture’s prediction for the 1level density. 1.
Some studies on arithmetical chaos in classical and quantum mechanics
 Internat. J. Modern Phys
, 1993
"... Several aspects of classical and quantum mechanics applied to a class of strongly chaotic systems are studied. The latter consists of single particles moving without external forces on surfaces of constant negative Gaussian curvature whose corresponding fundamental groups are supplied with an arithm ..."
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Cited by 7 (0 self)
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Several aspects of classical and quantum mechanics applied to a class of strongly chaotic systems are studied. The latter consists of single particles moving without external forces on surfaces of constant negative Gaussian curvature whose corresponding fundamental groups are supplied with an arithmetic structure. It is shown that the arithmetical features of the considered systems lead to exceptional properties of the corresponding spectra of lengths of closed geodesics (periodic orbits). The most significant one is an exponential growth of degeneracies in these geodesic length spectra. Furthermore, the arithmetical systems are distinguished by a structure that appears as a generalization of geometric symmetries. These pseudosymmetries occur in the quantization of the classical arithmetic systems as Hecke operators, which form an infinite algebra of selfadjoint operators commuting with the Hamiltonian. The statistical properties of quantum energies in the arithmetical systems have previously been identified as exceptional. They do not fit into the general scheme of random matrix theory. It is shown with the help of a simplified model for the spectral form factor
An orthogonal test of the LFunctions Ratios Conjecture
"... ABSTRACT. We test the predictions of the Lfunctions Ratios Conjecture for the family of cuspidal newforms of weight k and level N, with either k fixed and N → ∞ through the primes or N = 1 and k → ∞. We study the main and lower order terms in the 1level density. We provide evidence for the Ratios ..."
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Cited by 5 (3 self)
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ABSTRACT. We test the predictions of the Lfunctions Ratios Conjecture for the family of cuspidal newforms of weight k and level N, with either k fixed and N → ∞ through the primes or N = 1 and k → ∞. We study the main and lower order terms in the 1level density. We provide evidence for the Ratios Conjecture by computing and confirming its predictions up to a power savings in the family’s cardinality, at least for test functions whose Fourier transforms are supported in (−2, 2). We do this both for the weighted and unweighted 1level density (where in the weighted case we use the Petersson weights), thus showing that either formulation may be used. These two 1level densities differ by a term of size 1 / log(k 2 N). Finally, we show that there is another way of extending the sums arising in the Ratios Conjecture, leading to a different answer (although the answer is such a lower order term that it is hopeless to observe which is correct). 1.