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The primes contain arbitrarily long arithmetic progressions
 Ann. of Math
"... Abstract. We prove that there are arbitrarily long arithmetic progressions of primes. ..."
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Cited by 169 (27 self)
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Abstract. We prove that there are arbitrarily long arithmetic progressions of primes.
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 114 (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 90 (16 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 primes contain arbitrarily long polynomial progressions
 Acta Math
"... Abstract. We establish the existence of infinitely many polynomial progressions in the primes; more precisely, given any integervalued polynomials P1,..., Pk ∈ Z[m] in one unknown m with P1(0) =... = Pk(0) = 0 and any ε> 0, we show that there are infinitely many integers x, m with 1 ≤ m ≤ x ε ..."
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Cited by 32 (4 self)
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Abstract. We establish the existence of infinitely many polynomial progressions in the primes; more precisely, given any integervalued polynomials P1,..., Pk ∈ Z[m] in one unknown m with P1(0) =... = Pk(0) = 0 and any ε> 0, we show that there are infinitely many integers x, m with 1 ≤ m ≤ x ε such that x+P1(m),..., x+Pk(m) are simultaneously prime. The arguments are based on those in [18], which treated the linear case Pi = (i − 1)m and ε = 1; the main new features are a localization of the shift parameters (and the attendant Gowers norm objects) to both coarse and fine scales, the use of PET induction to linearize the polynomial averaging, and some elementary estimates for the number of points over finite fields in certain algebraic varieties. Contents
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 22 (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.
Quantum Unique Ergodicity For Eisenstein Series On PSL 2 (Z)\PSL 2 (R)
 Z)\PSL2(R), Ann. Inst. Fourier (Grenoble
, 1994
"... . In this paper we prove microlocal version of the equidistribution theorem for Wigner distributions associated to Eisenstein series on PSL 2 (Z)nPSL 2 (R). This generalizes a recent result of W. Luo and P. Sarnak who prove equidistribution for PSL 2 (Z)nH. The averaged versions of these results ha ..."
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Cited by 14 (0 self)
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. In this paper we prove microlocal version of the equidistribution theorem for Wigner distributions associated to Eisenstein series on PSL 2 (Z)nPSL 2 (R). This generalizes a recent result of W. Luo and P. Sarnak who prove equidistribution for PSL 2 (Z)nH. The averaged versions of these results have been proven by Zelditch for an arbitrary finitevolume surface, but our proof depends essentially on the presence of Hecke operators and works only for congruence subgroups of SL 2 (R). In the proof the key estimates come from applying Meurman's and Good's results on Lfunctions associated to holomorphic and Maass cusp forms. One also has to use classical transformation formulas for generalized hypergeometric functions of a unit argument. R' esum' e. Nous donnons la preuve d'une version microlocale d'un resultat de W. Luo et P. Sarnak concernant la r'epartition asymptotique des fonctions de Wigner associ'ees aux series d'Eisenstein sur PSL 2 (Z)nPSL 2 (R). La preuve utilise les op'erateur...
A STRENGTHENING OF THE NYMANBEURLING CRITERION FOR THE RIEMANN HYPOTHESIS, 2
, 2002
"... Let ρ(x) = x−[x], χ = χ(0,1). In L2(0, ∞) consider the subspace B generated by {ρaa ≥ 1} where ρa(x): = ρ () 1. By the Nymanax Beurling criterion the Riemann hypothesis is equivalent to the statement χ ∈ B. For some time it has been conjectured, and proved in the first version of this paper, post ..."
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Cited by 12 (2 self)
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Let ρ(x) = x−[x], χ = χ(0,1). In L2(0, ∞) consider the subspace B generated by {ρaa ≥ 1} where ρa(x): = ρ () 1. By the Nymanax Beurling criterion the Riemann hypothesis is equivalent to the statement χ ∈ B. For some time it has been conjectured, and proved in the first version of this paper, posted in arXiv:math.NT/0202141 v2, that the Riemann hypothesis is equivalent to the stronger statement that χ ∈ B nat where B nat is the much smaller subspace generated by {ρaa ∈ N}. This second version differs from the first in showing that under the Riemann hypothesis the distance between χ and − ∑ log a n a=1 µ(a)e−c order (log log n) −1/3.
The numbertheoretical spin chain and the Riemann zeroes
 Comm. Math. Phys
, 1998
"... Abstract It is an empirical observation that the Riemann zeta function can be well approximated in its critical strip using the NumberTheoretical Spin Chain. A proof of this would imply the Riemann Hypothesis. Here we relate that question to the one of spectral radii of a family of Markov chains. T ..."
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Cited by 9 (1 self)
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Abstract It is an empirical observation that the Riemann zeta function can be well approximated in its critical strip using the NumberTheoretical Spin Chain. A proof of this would imply the Riemann Hypothesis. Here we relate that question to the one of spectral radii of a family of Markov chains. This in turn leads to the question whether certain graphs are Ramanujan. The general idea is to explain the pseudorandom features of certain numbertheoretical functions by considering them as observables of a spin chain of statistical mechanics. In an Appendix we relate the free energy of that chain to the Lewis Equation of modular theory. 1 Introduction The Euler
Number Theory, Dynamical Systems and Statistical Mechanics
, 1998
"... We shortly review recent work interpreting the quotient ζ(s − 1)/ζ(s) of Riemann zeta functions as a dynamical zeta function. The corresponding interaction function (Fourier transform of the energy) has been shown to be ferromagnetic, i.e. positive. On the additive group we set inductively Gk: = (Z/ ..."
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Cited by 8 (2 self)
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We shortly review recent work interpreting the quotient ζ(s − 1)/ζ(s) of Riemann zeta functions as a dynamical zeta function. The corresponding interaction function (Fourier transform of the energy) has been shown to be ferromagnetic, i.e. positive. On the additive group we set inductively Gk: = (Z/2Z) k, with Z/2Z = ({0, 1}, +). h0: = 1, hk+1(σ, 0): = hk(σ) and hk+1(σ, 1): = hk(σ) + hk(1 − σ), (1) where σ = (σ1,..., σk) ∈ Gk and 1 − σ: = (1 − σ1,..., 1 − σk) is the inverted configuration. The sequences hk(σ) of integers, written in lexicographic order, coincide with the denominators of the modified Farey sequence. We now formally interpret σ ∈ Gk as a configuration of a spin chain with k spins and energy function Hk: = ln(hk). Thus we may interpret