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562
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 175 (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.
Linear equations in primes
 ANNALS OF MATHEMATICS
, 2006
"... Consider a system Ψ of nonconstant affinelinear forms ψ1,..., ψt: Z d → Z, no two of which are linearly dependent. Let N be a large integer, and let K ⊆ [−N, N] d be convex. A generalisation of a famous and difficult open conjecture of Hardy and Littlewood predicts an asymptotic, as N → ∞, for th ..."
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Cited by 79 (3 self)
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Consider a system Ψ of nonconstant affinelinear forms ψ1,..., ψt: Z d → Z, no two of which are linearly dependent. Let N be a large integer, and let K ⊆ [−N, N] d be convex. A generalisation of a famous and difficult open conjecture of Hardy and Littlewood predicts an asymptotic, as N → ∞, for the number of integer points n ∈ Z d ∩ K for which the integers ψ1(n),..., ψt(n) are simultaneously prime. This implies many other wellknown conjectures, such as the twin prime conjecture and the (weak) Goldbach conjecture. It also allows one to count the number of solutions in a convex range to any simultaneous linear system of equations, in which all unknowns are required to be prime. In this paper we (conditionally) verify this asymptotic under the assumption that no two of the affinelinear forms ψ1,..., ψt are affinely related; this excludes the important “binary ” cases such as the twin prime or Goldbach conjectures, but does allow one to count “nondegenerate ” configurations such as arithmetic progressions. Our result assumes two families of conjectures, which we term the inverse Gowersnorm conjecture (GI(s)) and the Möbius and nilsequences conjecture (MN(s)), where s ∈ {1, 2,...} is
Affine linear sieve, expanders, and sumproduct
"... This paper is concerned with the following general problem. For j = 1, 2,...,k let Aj be invertible integer coefficient polynomial maps of Z n to Z n (here n ≥ 1 and the inverses of Aj’s are assumed to be of the same type). Let Λ be the group generated by A1,...,Ak and ..."
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Cited by 42 (8 self)
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This paper is concerned with the following general problem. For j = 1, 2,...,k let Aj be invertible integer coefficient polynomial maps of Z n to Z n (here n ≥ 1 and the inverses of Aj’s are assumed to be of the same type). Let Λ be the group generated by A1,...,Ak and
Harald Cramér and the distribution of prime numbers
 Scandanavian Actuarial J
, 1995
"... “It is evident that the primes are randomly distributed but, unfortunately, we don’t know what ‘random ’ means. ” — R. C. Vaughan (February 1990). After the first world war, Cramér began studying the distribution of prime numbers, guided by Riesz and MittagLeffler. His works then, and later in the ..."
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Cited by 38 (2 self)
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“It is evident that the primes are randomly distributed but, unfortunately, we don’t know what ‘random ’ means. ” — R. C. Vaughan (February 1990). After the first world war, Cramér began studying the distribution of prime numbers, guided by Riesz and MittagLeffler. His works then, and later in the midthirties, have had a profound influence on the way mathematicians think about the distribution of prime numbers. In this article, we shall focus on how Cramér’s ideas have directed and motivated research ever since. One can only fully appreciate the significance of Cramér’s contributions by viewing his work in the appropriate historical context. We shall begin our discussion with the ideas of the ancient Greeks, Euclid and Eratosthenes. Then we leap in time to the nineteenth century, to the computations and heuristics of Legendre and Gauss, the extraordinarily analytic insights of Dirichlet and Riemann, and the crowning glory of these ideas, the proof the “Prime Number Theorem ” by Hadamard and de la Vallée Poussin in 1896. We pick up again in the 1920’s with the questions asked by Hardy and Littlewood,
The Möbius function is strongly orthogonal to nilsequence
 Ann. of Math
"... Abstract We show that the Möbius function µ(n) is strongly asymptotically orthogonal to any polynomial nilsequence (F (g(n)Γ)) n∈N . Here, G is a simplyconnected nilpotent Lie group with a discrete and cocompact subgroup Γ (so G/Γ is a nilmanifold ), g : Z → G is a polynomial sequence, and In part ..."
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Cited by 32 (1 self)
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Abstract We show that the Möbius function µ(n) is strongly asymptotically orthogonal to any polynomial nilsequence (F (g(n)Γ)) n∈N . Here, G is a simplyconnected nilpotent Lie group with a discrete and cocompact subgroup Γ (so G/Γ is a nilmanifold ), g : Z → G is a polynomial sequence, and In particular, this implies the Möbius and Nilsequence conjecture MN(s) from our earlier paper for every positive integer s. This is one of two major ingredients in our programme to establish a large number of cases of the generalised HardyLittlewood conjecture, which predicts how often a collection ψ1, . . . , ψt : Z d → Z of linear forms all take prime values. The proof is a relatively quick application of the results in our recent companion paper. We give some applications of our main theorem. We show, for example, that the Möbius function is uncorrelated with any bracket polynomial such as n √ 3 n √ 2 . We also obtain a result about the distribution of nilsequences (a n xΓ) n∈N as n ranges only over the primes.
Expander graphs in pure and applied mathematics
 Bull. Amer. Math. Soc. (N.S
"... Expander graphs are highly connected sparse finite graphs. They play an important role in computer science as basic building blocks for network constructions, error correcting codes, algorithms and more. In recent years they have started to play an increasing role also in pure mathematics: number th ..."
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Cited by 30 (3 self)
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Expander graphs are highly connected sparse finite graphs. They play an important role in computer science as basic building blocks for network constructions, error correcting codes, algorithms and more. In recent years they have started to play an increasing role also in pure mathematics: number theory, group theory, geometry and more. This expository article describes their constructions and various applications in pure and applied mathematics. This paper is based on notes prepared for the Colloquium Lectures at the
INTEGRAL MOMENTS OF LFUNCTIONS
, 2005
"... We give a newheuristic for all of the main terms in the integral moments of various families of primitive Lfunctions. The results agree with previous conjectures for the leading order terms. Our conjectures also have an almost identical form to exact expressions for the corresponding moments of the ..."
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Cited by 29 (12 self)
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We give a newheuristic for all of the main terms in the integral moments of various families of primitive Lfunctions. The results agree with previous conjectures for the leading order terms. Our conjectures also have an almost identical form to exact expressions for the corresponding moments of the characteristic polynomials of either unitary, orthogonal, or symplectic matrices, where the moments are de ned by the appropriate group averages. This lends support to the idea that arithmetical Lfunctions have a spectral interpretation, and that their value distributions can be modeled using Random Matrix Theory. Numerical examples show good agreement with our conjectures.
GENERALIZATION OF SELBERG’S 3/16 THEOREM AND AFFINE SIEVE
"... A celebrated theorem of Selberg [32] states that for congruence subgroups of SL2(Z) there are no exceptional eigenvalues below 3/16. We prove a generalization of Selberg’s theorem for infinite index “congruence” subgroups of SL2(Z). Consequently we obtain sharp upper ..."
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Cited by 25 (3 self)
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A celebrated theorem of Selberg [32] states that for congruence subgroups of SL2(Z) there are no exceptional eigenvalues below 3/16. We prove a generalization of Selberg’s theorem for infinite index “congruence” subgroups of SL2(Z). Consequently we obtain sharp upper
The large sieve, monodromy and zeta functions of curves
, 2005
"... We prove a large sieve statement for the average distribution of Frobenius conjugacy classes in arithmetic monodromy groups over finite fields. As a first application we prove a stronger version of a result of Chavdarov on the “generic” irreducibility of the numerator of the zeta functions in a fami ..."
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Cited by 23 (6 self)
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We prove a large sieve statement for the average distribution of Frobenius conjugacy classes in arithmetic monodromy groups over finite fields. As a first application we prove a stronger version of a result of Chavdarov on the “generic” irreducibility of the numerator of the zeta functions in a family of curves with large monodromy.