Results 1  10
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105
Matrix models for betaensembles
 J. Math. Phys
, 2002
"... This paper constructs tridiagonal random matrix models for general (β> 0) βHermite (Gaussian) and βLaguerre (Wishart) ensembles. These generalize the wellknown Gaussian and Wishart models for β = 1,2,4. Furthermore, in the cases of the βLaguerre ensembles, we eliminate the exponent quantizati ..."
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Cited by 102 (19 self)
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This paper constructs tridiagonal random matrix models for general (β> 0) βHermite (Gaussian) and βLaguerre (Wishart) ensembles. These generalize the wellknown Gaussian and Wishart models for β = 1,2,4. Furthermore, in the cases of the βLaguerre ensembles, we eliminate the exponent quantization present in the previously known models. We further discuss applications for the new matrix models, and present some open problems.
Probability laws related to the Jacobi theta and Riemann zeta functions, and Brownian excursions
 BULLETIN (NEW SERIES) OF THE AMERICAN MATHEMATICAL SOCIETY
, 2001
"... This paper reviews known results which connect Riemann’s integral representations of his zeta function, involving Jacobi’s theta function and its derivatives, to some particular probability laws governing sums of independent exponential variables. These laws are related to onedimensional Brownian ..."
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Cited by 70 (14 self)
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This paper reviews known results which connect Riemann’s integral representations of his zeta function, involving Jacobi’s theta function and its derivatives, to some particular probability laws governing sums of independent exponential variables. These laws are related to onedimensional Brownian motion and to higher dimensional Bessel processes. We present some characterizations of these probability laws, and some approximations of Riemann’s zeta function which are related to these laws.
Computational strategies for the Riemann zeta function
, 2000
"... We provide a compendium of evaluation methods for the Riemann zeta function, presenting formulae ranging from historical attempts to recently found convergent series to curious oddities old and new. We concentrate primarily on practical computational issues, such issues depending on the domain of th ..."
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Cited by 60 (11 self)
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We provide a compendium of evaluation methods for the Riemann zeta function, presenting formulae ranging from historical attempts to recently found convergent series to curious oddities old and new. We concentrate primarily on practical computational issues, such issues depending on the domain of the argument, the desired speed of computation, and the incidence of what we call “value recycling”.
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 52 (10 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.
Fast algorithms for multiple evaluations of the Riemann zeta function
 TRANS. AMER. MATH. SOC
, 1988
"... The best previously known algorithm for evaluating the Riemann zeta function, c,(a + it), with a bounded and t large to moderate accuracy (within ±t~c for some c> 0, say) was based on the RiemannSiegel formula and required on the order of fc1/2 operations for each value that was computed. New a ..."
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Cited by 52 (7 self)
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The best previously known algorithm for evaluating the Riemann zeta function, c,(a + it), with a bounded and t large to moderate accuracy (within ±t~c for some c> 0, say) was based on the RiemannSiegel formula and required on the order of fc1/2 operations for each value that was computed. New algorithms are presented in this paper which enable one to compute any single value of c(cr + it) with a fixed and T < t < T + Tlf2 to within ±t~c in 0(te) operations on numbers of O(logi) bits for any e> 0, for example, provided a precomputation involving 0(T1f2+e) operations and 0(T1f2+e) bits of storage is carried out beforehand. These algorithms lead to methods for numerically verifying the Riemann hypothesis for the first n zeros in what is expected to be 0(n1+s) operations (as opposed to about n3/2 operations for the previous method), as well as improved algorithms for the computation of various arithmetic functions, such as 7r(i). The new zeta function algorithms use the fast Fourier transform and a new method for the evaluation of certain rational functions. They can also be applied to the evaluation of Lfunctions, Epstein zeta functions, and other Dirichlet series.
Lowlying zeros of Lfunctions and random matrix theory
 Duke Math. J
, 2001
"... By looking at the average behavior (nlevel density) of the lowlying zeros of certain families of Lfunctions, we find evidence, as predicted by function field analogs, in favor of a spectral interpretation of the nontrivial zeros in terms of the classical compact groups. 1. ..."
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Cited by 45 (0 self)
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By looking at the average behavior (nlevel density) of the lowlying zeros of certain families of Lfunctions, we find evidence, as predicted by function field analogs, in favor of a spectral interpretation of the nontrivial zeros in terms of the classical compact groups. 1.
Evidence for a Spectral Interpretation of the Zeros of LFunctions
, 1998
"... By looking at the average behavior (nlevel density) of the low lying zeros of certain families of Lfunctions, we find evidence, as predicted by function field analogs, in favor of a spectral interpretation of the nontrivial zeros in terms of the classical compact groups. This is further supported ..."
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Cited by 45 (10 self)
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By looking at the average behavior (nlevel density) of the low lying zeros of certain families of Lfunctions, we find evidence, as predicted by function field analogs, in favor of a spectral interpretation of the nontrivial zeros in terms of the classical compact groups. This is further supported by numerical experiments for which an efficient algorithm to compute Lfunctions was developed and implemented. iii Acknowledgements When Mike Rubinstein woke up one morning he was shocked to discover that he was writing the acknowledgements to his thesis. After two screenplays, a 40000 word manifesto, and many fruitless attempts at making sushi, something resembling a detailed academic work has emerged for which he has people to thank. Peter Sarnak from Chebyshev's Bias to USp(1). For being a terrific advisor and teacher. For choosing problems suited to my talents and involving me in this great project to understand the zeros of Lfunctions. Zeev Rudnick and Andrew Oldyzko for many disc...
Orthogonal polynomial ensembles in probability theory
 Prob. Surv
, 2005
"... Abstract: We survey a number of models from physics, statistical mechanics, probability theory and combinatorics, which are each described in terms of an orthogonal polynomial ensemble. The most prominent example is apparently the Hermite ensemble, the eigenvalue distribution of the Gaussian Unitary ..."
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Cited by 32 (1 self)
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Abstract: We survey a number of models from physics, statistical mechanics, probability theory and combinatorics, which are each described in terms of an orthogonal polynomial ensemble. The most prominent example is apparently the Hermite ensemble, the eigenvalue distribution of the Gaussian Unitary Ensemble (GUE), and other wellknown ensembles known in random matrix theory like the Laguerre ensemble for the spectrum of Wishart matrices. In recent years, a number of further interesting models were found to lead to orthogonal polynomial ensembles, among which the corner growth model, directed last passage percolation, the PNG droplet, noncolliding random processes, the length of the longest increasing subsequence of a random permutation, and others. Much attention has been paid to universal classes of asymptotic behaviors of these models in the limit of large particle numbers, in particular the spacings between the particles and the fluctuation behavior of the largest particle. Computer simulations suggest that the connections go even farther
The effect of convolving families of Lfunctions on the underlying group symmetries
 JOURNAL OF NUMBER THEORY
, 2008
"... Let {FN} and {GM} be families of primitive automorphic Lfunctions for GLn(AQ) and GLm(AQ), respectively, such that, as N, M → ∞, the statistical behavior (1level density) of the lowlying zeros of Lfunctions in FN (resp., GM) agrees with that of the eigenvalues near 1 of matrices in G1 (resp., ..."
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Cited by 21 (16 self)
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Let {FN} and {GM} be families of primitive automorphic Lfunctions for GLn(AQ) and GLm(AQ), respectively, such that, as N, M → ∞, the statistical behavior (1level density) of the lowlying zeros of Lfunctions in FN (resp., GM) agrees with that of the eigenvalues near 1 of matrices in G1 (resp., G2) as the size of the matrices tend to infinity, where each Gi is one of the classical compact groups (unitary U, symplectic Sp, or orthogonal O, SO(even), SO(odd)). Assuming that the convolved families of Lfunctions FN × GM are automorphic, we study their 1level density. (We also study convolved families of the form f × GM for a fixed f.) Under natural assumptions on the families (which hold in many cases) we can associate to each family L of Lfunctions a symmetry constant cL equal to 0 (resp., 1 or −1) if the corresponding lowlying zero statistics agree with those of the unitary (resp., symplectic or orthogonal) group. Our main result is that cF×G = cF · cG: the symmetry type of the convolved family is the product of the symmetry types of the two families. A similar statement holds for the convolved families f × GM. We provide examples built from Dirichlet Lfunctions and holomorphic modular forms and their symmetric powers. An interesting special case is to convolve two families of elliptic curves with rank. In this case the symmetry group of the convolution is independent of the ranks, in accordance with the general principle of multiplicativity of the symmetry constants (but the ranks persist, before taking the limit N, M → ∞, as lowerorder terms)