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98
Zeroes of Zeta Functions and Symmetry
, 1999
"... Hilbert and Polya suggested that there might be a natural spectral interpretation of the zeroes of the Riemann Zeta function. While at the time there was little evidence for this, today the evidence is quite convincing. Firstly, there are the “function field” analogues, that is zeta functions of cur ..."
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Cited by 104 (2 self)
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Hilbert and Polya suggested that there might be a natural spectral interpretation of the zeroes of the Riemann Zeta function. While at the time there was little evidence for this, today the evidence is quite convincing. Firstly, there are the “function field” analogues, that is zeta functions of curves over finite fields and their generalizations. For these a spectral interpretation for their zeroes exists in terms of eigenvalues of Frobenius on cohomology. Secondly, the developments, both theoretical and numerical, on the local spacing distributions between the high zeroes of the zeta function and its generalizations give striking evidence for such a spectral connection. Moreover, the lowlying zeroes of various families of zeta functions follow laws for the eigenvalue distributions of members of the classical groups. In this paper we review these developments. In order to present the material fluently, we do not proceed in chronological order of discovery. Also, in concentrating entirely on the subject matter of the title, we are ignoring the standard body of important work that has been done on the zeta function and Lfunctions.
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
Modularity Of The RankinSelberg LSeries, And Multiplicity One For SL(2)
"... Contents 1. Introduction 1 2. Notations and Preliminaries 5 3. Construction of # : A(GL(2)) A(GL(2)) # A(GL(4)) 8 3.1. Relevant objects and the strategy 9 3.2. Weak to strong lifting, and the cuspidality criterion 13 3.3. Triple product Lfunctions: local factors and holomorphy 15 3.4. Boundednes ..."
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Cited by 57 (13 self)
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Contents 1. Introduction 1 2. Notations and Preliminaries 5 3. Construction of # : A(GL(2)) A(GL(2)) # A(GL(4)) 8 3.1. Relevant objects and the strategy 9 3.2. Weak to strong lifting, and the cuspidality criterion 13 3.3. Triple product Lfunctions: local factors and holomorphy 15 3.4. Boundedness in vertical strips 18 3.5. Modularity in the good case 30 3.6. A descent criterion 32 3.7. Modularity in the general case 35 4. Applications 37 4.1. A multiplicity one theorem for SL(2) 37 4.2. Some new functional equations 40 4.3. Root numbers and representations of orthogonal type 42 4.4. Triple product Lfunctions revisited 44 4.5. The Tate conjecture for 4fold products of modular curves 47 Bibliography 52 1. Introduction Let f, g be primitive cusp forms, holomorphic or otherwise, on
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 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.
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 33 (7 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...
Autocorrelation of random matrix polynomials
 COMMUN. MATH. PHYS
, 2003
"... We calculate the autocorrelation functions (or shifted moments) of the characteristic polynomials of matrices drawn uniformly with respect to Haar measure from the groups U(N), O(2N) and USp(2N). In each case the result can be expressed in three equivalent forms: as a determinant sum (and hence in t ..."
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Cited by 32 (17 self)
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We calculate the autocorrelation functions (or shifted moments) of the characteristic polynomials of matrices drawn uniformly with respect to Haar measure from the groups U(N), O(2N) and USp(2N). In each case the result can be expressed in three equivalent forms: as a determinant sum (and hence in terms of symmetric polynomials), as a combinatorial sum, and as a multiple contour integral. These formulae are analogous to those previously obtained for the Gaussian ensembles of Random Matrix Theory, but in this case are identities for any size of matrix, rather than largematrix asymptotic approximations. They also mirror exactly the autocorrelation formulae conjectured to hold for Lfunctions in a companion paper. This then provides further evidence in support of the connection between Random Matrix Theory and the theory of Lfunctions.
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 30 (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.
Universality for mathematical and physical systems
, 2006
"... Abstract. All physical systems in equilibrium obey the laws of thermodynamics. In other words, whatever the precise nature of the interaction between the atoms and molecules at the microscopic level, at the macroscopic level, physical systems exhibit universal behavior in the sense that they are all ..."
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Cited by 21 (0 self)
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Abstract. All physical systems in equilibrium obey the laws of thermodynamics. In other words, whatever the precise nature of the interaction between the atoms and molecules at the microscopic level, at the macroscopic level, physical systems exhibit universal behavior in the sense that they are all governed by the same laws and formulae of thermodynamics. In this paper we describe some recent history of universality ideas in physics starting with Wigner’s model for the scattering of neutrons off large nuclei and show how these ideas have led mathematicians to investigate universal behavior for a variety of mathematical systems. This is true not only for systems which have a physical origin, but also for systems which arise in a purely mathematical context such as the Riemann hypothesis, and a version of the card game solitaire called patience sorting. 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 19 (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.