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139
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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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.
Zerofree regions for Dirichlet Lfunctions and the least prime in an arithmetic progression
 Proc. Lond. Math. Soc
, 1992
"... The classical theorem of Dirichlet states that any arithmetic progression a(mod q) in which a and q are relatively prime contains infinitely many prime numbers. A natural question to ask is then, how big is the first such prime, P (a, q) say? In one direction we have trivially ..."
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Cited by 46 (0 self)
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The classical theorem of Dirichlet states that any arithmetic progression a(mod q) in which a and q are relatively prime contains infinitely many prime numbers. A natural question to ask is then, how big is the first such prime, P (a, q) say? In one direction we have trivially
Practical ZeroKnowledge Proofs: Giving Hints and Using Deficiencies
 JOURNAL OF CRYPTOLOGY
, 1994
"... New zeroknowledge proofs are given for some numbertheoretic problems. All of the problems are in NP, but the proofs given here are much more efficient than the previously known proofs. In addition, these proofs do not require the prover to be superpolynomial in power. A probabilistic polynomial t ..."
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Cited by 32 (0 self)
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New zeroknowledge proofs are given for some numbertheoretic problems. All of the problems are in NP, but the proofs given here are much more efficient than the previously known proofs. In addition, these proofs do not require the prover to be superpolynomial in power. A probabilistic polynomial time prover with the appropriate trapdoor knowledge is sufficient. The proofs are perfect or statistical zeroknowledge in all cases except one.
The cubic moment of central values of automorphic Lfunctions
 Ann. of Math
, 2000
"... 2. A review of classical modular forms 3. A review of Maass forms 4. Hecke Lfunctions ..."
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Cited by 24 (1 self)
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2. A review of classical modular forms 3. A review of Maass forms 4. Hecke Lfunctions
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 23 (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,
Ramanujan’s ternary quadratic form
 Invent. Math
, 1997
"... In [R], S. Ramanujan investigated the representation of integers by positive definite quadratic forms, and the ternary form (1) φ1(x, y, z): = x2 + y2 + 10z2 was of particular interest to him. This form is in a genus consisting of two classes, and ..."
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In [R], S. Ramanujan investigated the representation of integers by positive definite quadratic forms, and the ternary form (1) φ1(x, y, z): = x2 + y2 + 10z2 was of particular interest to him. This form is in a genus consisting of two classes, and
Large Character Sums
 CHARACTERS AND THE POLYAVINOGRADOV THEOREM 29
"... A central problem in analytic number theory is to gain an understanding of character sums χ(n), n≤x ..."
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A central problem in analytic number theory is to gain an understanding of character sums χ(n), n≤x
Homomorphic PublicKey Cryptosystems and Encrypting Boolean Circuits
, 2003
"... In this paper homomorphic cryptosystems are designed for the first time over any finite group. Applying Barrington's construction we produce for any boolean circuit of the logarithmic depth its encrypted simulation of a polynomial size over an appropriate finitely generated group. ..."
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In this paper homomorphic cryptosystems are designed for the first time over any finite group. Applying Barrington's construction we produce for any boolean circuit of the logarithmic depth its encrypted simulation of a polynomial size over an appropriate finitely generated group.