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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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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 s ..."
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Cited by 28 (5 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 23 (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.
Surpassing the Ratios Conjecture in the 1level density of Dirichlet Lfunctions
 ALGEBRA & NUMBER THEORY
, 2012
"... ..."
Large gaps between consecutive zeros of the Riemann zetafunction
, 2009
"... 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 8 (3 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.
On the distribution of pseudopowers
"... Introduced by Kraitchik and Lehmer, an xpseudosquare is a positive integer n 1 (mod 8) that is a quadratic residue for each odd prime p x, yet is not a square. We use bounds of character sums to prove that pseudosquares are equidistributed in fairly short intervals. An xpseudopower to base g i ..."
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Introduced by Kraitchik and Lehmer, an xpseudosquare is a positive integer n 1 (mod 8) that is a quadratic residue for each odd prime p x, yet is not a square. We use bounds of character sums to prove that pseudosquares are equidistributed in fairly short intervals. An xpseudopower to base g is a positive integer which is not a power of g yet is so modulo p for all primes p x. It is conjectured by Bach, Lukes, Shallit, and Williams that the least such number is at most exp(agx = log x) for a suitable constant ag. A bound of exp(agx log log x = log x) is proved conditionally on the Riemann Hypothesis for Dedekind zeta functions, thus improving on a recent conditional exponential bound of Konyagin and the present authors. We also give a GRHconditional equidistribution result for pseudopowers that is analogous to our unconditional result for pseudosquares. 1
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 o ..."
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Cited by 7 (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 ...