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33
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.
Semiclassical Form Factor for Chaotic Systems With Spin 1/2
, 1999
"... . We study the properties of the twopoint spectral form factor for classically chaotic systems with spin 1/2 in the semiclassical limit, with a suitable semiclassical trace formula as our principal tool. To this end we introduce a regularized form factor and discuss the limit in which the socalled ..."
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Cited by 20 (16 self)
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. We study the properties of the twopoint spectral form factor for classically chaotic systems with spin 1/2 in the semiclassical limit, with a suitable semiclassical trace formula as our principal tool. To this end we introduce a regularized form factor and discuss the limit in which the socalled diagonal approximation can be recovered. The incorporation of the spin contribution to the trace formula requires an appropriate variant of the equidistribution principle of long periodic orbits as well as the notion of a skew product of the classical translational and spin dynamics. Provided this skew product is mixing, we show that generically the diagonal approximation of the form factor coincides with the respective predictions from random matrix theory. PACS numbers: 03.65.Sq, 05.45.Mt k Email address: bol@physik.uniulm.de  Email address: kep@physik.uniulm.de + Address after 1 October 1999: Abteilung Theoretische Physik, Universitat Ulm, AlbertEinsteinAllee 11, D89069 Ulm, G...
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
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On the spacing distribution of the Riemann zeros : corrections to the asymptotic result
 J. Phys. A : Math. Gen
"... It has been conjectured that the statistical properties of zeros of the Riemann zeta function near z = 1/2 + iE tend, as E → ∞, to the distribution of eigenvalues of large random matrices from the Unitary Ensemble. At finite E numerical results show that the nearestneighbour spacing distribution pr ..."
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Cited by 11 (1 self)
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It has been conjectured that the statistical properties of zeros of the Riemann zeta function near z = 1/2 + iE tend, as E → ∞, to the distribution of eigenvalues of large random matrices from the Unitary Ensemble. At finite E numerical results show that the nearestneighbour spacing distribution presents deviations with respect to the conjectured asymptotic form. We give here arguments indicating that to leading order these deviations are the same as those of unitary random matrices of finite dimension Neff = log(E/2π) / √ 12Λ, where Λ = 1.57314... is a well defined constant. 1
Lower order terms in the 1level density for families of holomorphic cuspidal newforms
"... ABSTRACT. The KatzSarnak density conjecture states that, in the limit as the conductors tend to infinity, the behavior of normalized zeros near the central point of families of Lfunctions agree with the N → ∞ scaling limits of eigenvalues near 1 of subgroups of U(N). Evidence for this has been fo ..."
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Cited by 9 (6 self)
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ABSTRACT. The KatzSarnak density conjecture states that, in the limit as the conductors tend to infinity, the behavior of normalized zeros near the central point of families of Lfunctions agree with the N → ∞ scaling limits of eigenvalues near 1 of subgroups of U(N). Evidence for this has been found for many families by studying the nlevel densities; for suitably restricted test functions the main terms agree with random matrix theory. In particular, all oneparameter families of elliptic curves with rank r over Q(T) and the same distribution of signs of functional equations have the same limiting behavior. We break this universality and find family dependent lower order correction terms in many cases; these lower order terms have applications ranging from excess rank to modeling the behavior of zeros near the central point, and depend on the arithmetic of the family. We derive an alternate form of the explicit formula for GL(2) Lfunctions which simplifies comparisons, replacing sums over powers of Satake parameters by sums of the moments of the Fourier coefficients λf(p). Our formula highlights the differences that we expect to exist from families whose Fourier coefficients obey different laws (for example, we expect SatoTate to hold only for nonCM families of elliptic curves). Further, by the work of Rosen and Silverman we expect lower order biases to the Fourier coefficients in families of elliptic curves with rank over Q(T); these biases can be seen in our expansions. We analyze several families of elliptic curves and see different lower order corrections, depending on whether or not the family has complex multiplication, a forced torsion point, or nonzero rank over Q(T). 1.
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.
Triple correlation of the Riemann zeros
"... We use the conjecture of Conrey, Farmer and Zirnbauer for averages of ratios of the Riemann zeta function [11] to calculate all the lower order terms of the triple correlation function of the Riemann zeros. A previous approach was suggested by Bogomolny and Keating [6] taking inspiration from semi ..."
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Cited by 4 (2 self)
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We use the conjecture of Conrey, Farmer and Zirnbauer for averages of ratios of the Riemann zeta function [11] to calculate all the lower order terms of the triple correlation function of the Riemann zeros. A previous approach was suggested by Bogomolny and Keating [6] taking inspiration from semiclassical methods. At that point they did not write out the answer explicitly, so we do that here, illustrating that by our method all the lower order terms down to the constant can be calculated rigourously if one assumes the ratios conjecture of Conrey, Farmer and Zirnbauer. Bogomolny and Keating [4] returned to their previous results simultaneously with this current work, and have written out the full expression. The result presented in this paper agrees precisely with their formula, as well as with our numerical computations, which we include here. We also include an alternate proof of the triple correlation of eigenvalues from random U(N) matrices which follows a nearly identical method to that for the Riemann zeros, but is based on