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36
Linear statistics of lowlying zeros of L–functions”, (preprint
"... Abstract. We consider linear statistics of the scaled zeros of Dirichlet L– functions, and show that the first few moments converge to the Gaussian moments. The number of Gaussian moments depends on the particular statistic considered. The same phenomenon is found in Random Matrix Theory, where we c ..."
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Cited by 21 (4 self)
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Abstract. We consider linear statistics of the scaled zeros of Dirichlet L– functions, and show that the first few moments converge to the Gaussian moments. The number of Gaussian moments depends on the particular statistic considered. The same phenomenon is found in Random Matrix Theory, where we consider linear statistics of scaled eigenphases for matrices in the unitary group. In that case the higher moments are no longer Gaussian. We conjecture that this also happens for Dirichlet L–functions. 1.
Rational Approximation To Algebraic Numbers Of Small Height: The Diophantine Equation ...
 1, J. Reine Angew. Math
"... Following an approach originally due to Mahler and sharpened by Chudnovsky, we develop an explicit version of the multidimensional "hypergeometric method" for rational and algebraic approximation to algebraic numbers. Consequently, if a; b and n are given positive integers with n 3, we show that t ..."
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Cited by 18 (2 self)
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Following an approach originally due to Mahler and sharpened by Chudnovsky, we develop an explicit version of the multidimensional "hypergeometric method" for rational and algebraic approximation to algebraic numbers. Consequently, if a; b and n are given positive integers with n 3, we show that the equation of the title possesses at most one solution in positive integers x; y. Further results on Diophantine equations are also presented. The proofs are based upon explicit Pad'e approximations to systems of binomial functions, together with new Chebyshevlike estimates for primes in arithmetic progressions and a variety of computational techniques. 1.
On quantum ergodicity for linear maps of the torus
 COMM. MATH. PHYS
, 1999
"... We prove a strong version of quantum ergodicity for linear hyperbolic maps of the torus (“cat maps”). We show that there is a density one sequence of integers so that as N tends to infinity along this sequence, all eigenfunctions of the quantum propagator at inverse Planck constant N are uniformly ..."
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Cited by 12 (3 self)
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We prove a strong version of quantum ergodicity for linear hyperbolic maps of the torus (“cat maps”). We show that there is a density one sequence of integers so that as N tends to infinity along this sequence, all eigenfunctions of the quantum propagator at inverse Planck constant N are uniformly distributed. A key step in the argument is to show that for a hyperbolic matrix in the modular group, there is a density one sequence of integers N for which its order (or period) modulo N is somewhat larger than √ N.
On the Size of the First Factor of the Class Number of a Cyclotomic Field
, 1990
"... We show that Kummer's conjectured asymptotic estimate for the size of the first factor of the class number of a cyclotomic field is untrue under the assumption of two wellknown and widely believed conjectures of analytic number theory. 1. Introduction In 1850 Kummer [13] published a review of the ..."
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Cited by 6 (2 self)
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We show that Kummer's conjectured asymptotic estimate for the size of the first factor of the class number of a cyclotomic field is untrue under the assumption of two wellknown and widely believed conjectures of analytic number theory. 1. Introduction In 1850 Kummer [13] published a review of the main results that he and others had discovered about cyclotomic fields. In this elegant report he claimed that he had found an explicit "law for the asymptotic growth" of h 1 (p), the socalled first factor of the class number of the cyclotomic field, and would provide a proof elsewhere. This proof never appeared and we believe that Kummer's claim is incorrect. More precisely, let p denote any odd prime, let h(p) be the class number of the cyclotomic field Q(i p ) (where i p is a primitive pth root of unity) and h 2 (p) be the class number of the real subfield Q(i p +i \Gamma1 p ). Kummer proved that the ratio h 1 (p) = h(p)=h 2 (p) is an integer which he called the first factor of the ...
The Lucas–Pratt primality tree
 Math. Comp
"... Abstract. In 1876, E. Lucas showed that a quick proof of primality for a prime p could be attained through the prime factorization of p − 1 and a primitive root for p. V. Pratt’s proof that PRIMES is in NP, done via Lucas’s theorem, showed that a certificate of primality for a prime p could be obtai ..."
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Cited by 4 (0 self)
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Abstract. In 1876, E. Lucas showed that a quick proof of primality for a prime p could be attained through the prime factorization of p − 1 and a primitive root for p. V. Pratt’s proof that PRIMES is in NP, done via Lucas’s theorem, showed that a certificate of primality for a prime p could be obtained in O(log 2 p) modular multiplications with integers at most p. We show that for all constants C ∈ R, the number of modular multiplications necessary to obtain this certificate is greater than C log p for a set of primes p with relative asymptotic density 1. 1.
On the ternary Goldbach problem with primes in independent arithmetic progressions
 Acta Math. Hungar
"... For A,ε> 0 and any sufficiently large odd n we show that for almost all k ≤ R: = n 1/5−ε there exists a representation n = p1 + p2 + p3 with primes pi ≡ bi mod k for almost all admissible triplets b1,b2,b3 of reduced residues mod k. ..."
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Cited by 2 (0 self)
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For A,ε> 0 and any sufficiently large odd n we show that for almost all k ≤ R: = n 1/5−ε there exists a representation n = p1 + p2 + p3 with primes pi ≡ bi mod k for almost all admissible triplets b1,b2,b3 of reduced residues mod k.
Large gaps between consecutive zeros of the Riemann zetafunction, preprint
"... Combining the mollifiers, we exhibit other choices of coefficients that improve the results on large gaps between the zeros of the Riemann zetafunction. Precisely, assuming the Generalized Riemann Hypothesis (GRH), we show that there exist infinitely many consecutive gaps greater than 3.0155 times ..."
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Cited by 2 (1 self)
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Combining the mollifiers, we exhibit other choices of coefficients that improve the results on large gaps between the zeros of the Riemann zetafunction. Precisely, assuming the Generalized Riemann Hypothesis (GRH), we show that there exist infinitely many consecutive gaps greater than 3.0155 times the average spacing. 1.
Explicit upper bounds for exponential sums over primes
 Math. Comp
"... Dedicated to the memory of Chen Jing Run Abstract. We give explicit upper bounds for linear trigonometric sums over primes. 1. ..."
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Cited by 1 (0 self)
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Dedicated to the memory of Chen Jing Run Abstract. We give explicit upper bounds for linear trigonometric sums over primes. 1.
DIVISORS OF SHIFTED PRIMES
"... Abstract. We bound from below the number of shifted primes p+s ≤ x that have a divisor in a given interval (y, z]. Kevin Ford has obtained upper bounds of the expected order of magnitude on this quantity as well as lower bounds in a special case of the parameters y and z. We supply here the correspo ..."
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Cited by 1 (1 self)
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Abstract. We bound from below the number of shifted primes p+s ≤ x that have a divisor in a given interval (y, z]. Kevin Ford has obtained upper bounds of the expected order of magnitude on this quantity as well as lower bounds in a special case of the parameters y and z. We supply here the corresponding lower bounds in a broad range of the parameters y and z. As expected, these bounds depend heavily on our knowledge about primes in arithmetic progressions. As an application of these bounds, we determine the number of shifted primes that appear in a multiplication table up to multiplicative constants. 1.