Results 1  10
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27
On the characteristic polynomial of a random unitary matrix
 Comm. Math. Phys
, 2001
"... Abstract: We present a range of fluctuation and large deviations results for the logarithm of the characteristic√polynomial Z of a random N × N unitary matrix, as N →∞. First 12 we show that ln Z / ln N, evaluated at a finite set of distinct points, is asymptotically a collection of i.i.d. complex n ..."
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Cited by 43 (11 self)
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Abstract: We present a range of fluctuation and large deviations results for the logarithm of the characteristic√polynomial Z of a random N × N unitary matrix, as N →∞. First 12 we show that ln Z / ln N, evaluated at a finite set of distinct points, is asymptotically a collection of i.i.d. complex normal random variables. This leads to a refinement of a recent central limit theorem due to Keating and Snaith, and also explains the covariance structure of the eigenvalue counting function. Next we obtain a central limit theorem for ln Z in a Sobolev space of generalised functions on the unit circle. In this limiting regime, lowerorder terms which reflect the global covariance structure are no longer negligible and feature in the covariance structure of the limiting Gaussian measure. Large deviations results for ln Z/A, evaluated at a finite set of distinct points, can be obtained for √ ln N ≪ A ≪ ln N. For higherorder scalings we obtain large deviations results for ln Z/A evaluated at a single point. There is a phase transition at A = ln N (which only applies to negative deviations of the real part) reflecting a switch from global to local conspiracy.
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 42 (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 matrix theory and the derivative of the Riemann zeta function
, 2000
"... Random matrix theory (RMT) is used to model the asymptotics of the discrete moments of the derivative of the Riemann zeta function, ? (s), evaluated at the complex zeros + iγn, using the methods introduced by Keating and Snaith in [14]. We also discuss the probability distribution of ln ? ´(1/2 + ..."
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Cited by 34 (7 self)
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Random matrix theory (RMT) is used to model the asymptotics of the discrete moments of the derivative of the Riemann zeta function, ? (s), evaluated at the complex zeros + iγn, using the methods introduced by Keating and Snaith in [14]. We also discuss the probability distribution of ln ? ´(1/2 + iγn), proving the central limit theorem for the corresponding random matrix distribution and analysing its large deviations.
Numerical computations concerning the ERH
 Math. Comp
, 1993
"... Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at ..."
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Cited by 25 (1 self)
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Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at
A hybrid EulerHadamard product formula for the Riemann zeta function
, 2005
"... We use a smoothed version of the explicit formula to find an approximation to the Riemann zeta function as a product over its nontrivial zeros multiplied by a product over the primes. We model the first product by characteristic polynomials of random matrices. This provides a statistical model of ..."
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Cited by 9 (2 self)
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We use a smoothed version of the explicit formula to find an approximation to the Riemann zeta function as a product over its nontrivial zeros multiplied by a product over the primes. We model the first product by characteristic polynomials of random matrices. This provides a statistical model of the zeta function that involves the primes in a natural way. We then employ the model in a heuristic calculation of the moments of the modulus of the zeta function on the critical line. This calculation illuminates recent conjectures for these moments based on connections with random matrix theory.
A multipole method for SchwarzChristoffel mapping of polygons with thousands of sides
 SIAM J. Sci. Comput
"... Abstract. A method is presented for the computation of Schwarz–Christoffel maps to polygons with tens of thousands of vertices. Previously published algorithms have CPU time estimates of the order O(N 3) for the computation of a conformal map of a polygon with N vertices. This has been reduced to O( ..."
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Cited by 7 (2 self)
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Abstract. A method is presented for the computation of Schwarz–Christoffel maps to polygons with tens of thousands of vertices. Previously published algorithms have CPU time estimates of the order O(N 3) for the computation of a conformal map of a polygon with N vertices. This has been reduced to O(N log N) by the use of the fast multipole method and Davis’s method for solving the parameter problem. The method is illustrated by a number of examples, the largest of which has N ≈ 2 × 10 5. Key words. snowflake conformal mapping, fast multipole method, Schwarz–Christoffel mapping, Koch
A HYBRID EULERHADAMARD PRODUCT FOR THE RIEMANN ZETA FUNCTION
"... We use a smoothed version of the explicit formula to find an accurate pointwise approximation to the Riemann zeta function as a product over its nontrivial zeros multiplied by a product over the primes. We model the first product by characteristic polynomials of random matrices. This provides a stat ..."
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Cited by 7 (4 self)
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We use a smoothed version of the explicit formula to find an accurate pointwise approximation to the Riemann zeta function as a product over its nontrivial zeros multiplied by a product over the primes. We model the first product by characteristic polynomials of random matrices. This provides a statistical model of the zeta function which involves the primes in a natural way. We then employ the model in a heuristic calculation of the moments of the modulus of the zeta function on the critical line. For the second and fourth moments, we establish all of the steps in our approach rigorously. This calculation illuminates recent conjectures for these moments based
SUPERCOMPUTERS AND THE RIEMANN ZETA FUNCTION
"... The Riemann Hypothesis, which specifies the location of zeros ofthe Riemann zeta function, and thus describes the behavior of primes, is one of the most famous unsolved problems inmathematics, and extensive efforts have been made over more than a century to check it numerically for large sets of c ..."
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Cited by 6 (0 self)
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The Riemann Hypothesis, which specifies the location of zeros ofthe Riemann zeta function, and thus describes the behavior of primes, is one of the most famous unsolved problems inmathematics, and extensive efforts have been made over more than a century to check it numerically for large sets of cases. Recently a new algorithm, invented by the speaker and A. Schönhage, has been implemented, and used to compute over 175 million zeros near zero number 10^20. The new algorithm turned out to be over 5 orders of magnitude faster than older methods. The crucial ingredients in it are a rational function evaluation method similar to the GreengardRokhlin gravitational potential evaluation algorithm, the FFT, andbandlimited function interpolation. While the only present implementation is on a Cray, the algorithm can easily be parallelized.
Linear statistics for zeros of Riemann’s zeta function
 I
"... Abstract. We consider a smooth counting function of the scaled zeros of the Riemann zeta function, around height T. We show that the first few moments tend to the Gaussian moments, with the exact number depending on the statistic considered. 1. ..."
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Cited by 5 (2 self)
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Abstract. We consider a smooth counting function of the scaled zeros of the Riemann zeta function, around height T. We show that the first few moments tend to the Gaussian moments, with the exact number depending on the statistic considered. 1.