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Computing discrete logarithms in real quadratic congruence function fields of large genus
 Math. Comp
, 1999
"... Abstract. The discrete logarithm problem in various finite abelian groups is the basis for some well known public key cryptosystems. Recently, real quadratic congruence function fields were used to construct a public key distribution system. The security of this public key system is based on the dif ..."
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Cited by 36 (8 self)
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Abstract. The discrete logarithm problem in various finite abelian groups is the basis for some well known public key cryptosystems. Recently, real quadratic congruence function fields were used to construct a public key distribution system. The security of this public key system is based on the difficulty of a discrete logarithm problem in these fields. In this paper, we present a probabilistic algorithm with subexponential running time that computes such discrete logarithms in real quadratic congruence function fields of sufficiently large genus. This algorithm is a generalization of similar algorithms for real quadratic number fields. 1.
Evidence for a Spectral Interpretation of the Zeros of LFunctions
, 1998
"... By looking at the average behavior (nlevel density) of the low lying zeros of certain families of Lfunctions, we find evidence, as predicted by function field analogs, in favor of a spectral interpretation of the nontrivial zeros in terms of the classical compact groups. This is further supported ..."
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Cited by 33 (7 self)
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By looking at the average behavior (nlevel density) of the low lying zeros of certain families of Lfunctions, we find evidence, as predicted by function field analogs, in favor of a spectral interpretation of the nontrivial zeros in terms of the classical compact groups. This is further supported by numerical experiments for which an efficient algorithm to compute Lfunctions was developed and implemented. iii Acknowledgements When Mike Rubinstein woke up one morning he was shocked to discover that he was writing the acknowledgements to his thesis. After two screenplays, a 40000 word manifesto, and many fruitless attempts at making sushi, something resembling a detailed academic work has emerged for which he has people to thank. Peter Sarnak from Chebyshev's Bias to USp(1). For being a terrific advisor and teacher. For choosing problems suited to my talents and involving me in this great project to understand the zeros of Lfunctions. Zeev Rudnick and Andrew Oldyzko for many disc...
Lowlying zeros of Lfunctions and random matrix theory
 Duke Math. J
, 2001
"... By looking at the average behavior (nlevel density) of the lowlying zeros of certain families of Lfunctions, we find evidence, as predicted by function field analogs, in favor of a spectral interpretation of the nontrivial zeros in terms of the classical compact groups. 1. ..."
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Cited by 31 (0 self)
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By looking at the average behavior (nlevel density) of the lowlying zeros of certain families of Lfunctions, we find evidence, as predicted by function field analogs, in favor of a spectral interpretation of the nontrivial zeros in terms of the classical compact groups. 1.
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 20 (1 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,
Cyclotomic integers and finite geometry
 J. Amer. Math. Soc
, 1999
"... The most powerful method for the study of finite geometries with regular or quasiregular automorphism groups G is to translate their definition into an equation over the integral group ring Z[G] and to investigate this equation by applying complex representations of G. For the definitions and the ba ..."
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Cited by 15 (1 self)
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The most powerful method for the study of finite geometries with regular or quasiregular automorphism groups G is to translate their definition into an equation over the integral group ring Z[G] and to investigate this equation by applying complex representations of G. For the definitions and the basic facts, see Section 2.
Comments on search procedures for primitive roots
 Math.Comp.66
, 1997
"... Abstract. Let p be an odd prime. Assuming the Extended Riemann Hypothesis, we show how to construct O((log p) 4 (log log p) −3) residues modulo p, one of which must be a primitive root, in deterministic polynomial time. Granting some wellknown character sum bounds, the proof is elementary, leading ..."
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Cited by 10 (0 self)
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Abstract. Let p be an odd prime. Assuming the Extended Riemann Hypothesis, we show how to construct O((log p) 4 (log log p) −3) residues modulo p, one of which must be a primitive root, in deterministic polynomial time. Granting some wellknown character sum bounds, the proof is elementary, leading to an explicit algorithm. 1.
A NEW BOUND FOR THE SMALLEST x WITH π(x)> li(x)
, 1999
"... Abstract. Let π(x) denote the number of primes ≤ x and let li(x) denotethe usual integral logarithm of x. We prove that there are at least 10153 integer values of x in the vicinity of 1.39822 × 10316 with π(x)> li(x). This improves earlier bounds of Skewes, Lehman, and te Riele. We also plot more th ..."
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Cited by 9 (0 self)
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Abstract. Let π(x) denote the number of primes ≤ x and let li(x) denotethe usual integral logarithm of x. We prove that there are at least 10153 integer values of x in the vicinity of 1.39822 × 10316 with π(x)> li(x). This improves earlier bounds of Skewes, Lehman, and te Riele. We also plot more than 10000 values of π(x) − li(x) in four different regions, including the regions discovered by Lehman, te Riele, and the authors of this paper, and a more distant region in the vicinity of 1.617 × 109608,whereπ(x) appears to exceed li(x) bymore than.18x 1 2 / log x. The plots strongly suggest, although upper bounds derived to date for li(x) − π(x) are not sufficient for a proof, that π(x) exceeds li(x) for at least 10311 integers in the vicinity of 1.398 × 10316. If it is possible to improve our bound for π(x) − li(x) by finding a sign change before 10316,our first plot clearly delineates the potential candidates. Finally, we compute the logarithmic density of li(x) − π(x) and find that as x departs from the region in the vicinity of 1.62 × 109608, the density is 1 − 2.7 × 10−7 =.99999973, and that it varies from this by no more than 9 × 10−8 over the next 1030000 integers. This should be compared to Rubinstein and Sarnak.
A Local Limit Theorem for Closed Geodesics and Homology
 Department of Mathematics, University of Manchester, Oxford Road, Manchester
, 2001
"... In this paper, we study the distribution of closed geodesics on a compact negatively curved manifold. We concentrate on geodesics lying in a prescribed homology class and, under certain conditions, obtain a local limit theorem to describe the asymptotic behaviour of the associated counting function ..."
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Cited by 7 (2 self)
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In this paper, we study the distribution of closed geodesics on a compact negatively curved manifold. We concentrate on geodesics lying in a prescribed homology class and, under certain conditions, obtain a local limit theorem to describe the asymptotic behaviour of the associated counting function as the homology class varies. 0.
Error Terms for Closed Orbits of Hyperbolic Flows
 Syst
"... this paper we shall consider error terms in estimates for the number of closed orbits for a large class of C 1 flows OE t : M ! M , restricted to a hyperbolic set . Let (T ) be the number of closed orbits of least period at most T ? 0. It is well known that h = lim T!+1 1 T log (T ); where h ..."
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Cited by 6 (4 self)
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this paper we shall consider error terms in estimates for the number of closed orbits for a large class of C 1 flows OE t : M ! M , restricted to a hyperbolic set . Let (T ) be the number of closed orbits of least period at most T ? 0. It is well known that h = lim T!+1 1 T log (T ); where h ? 0 denotes the topological entropy of the flow. It was shown by Parry and the first author in [7] that if OE is a weakmixing Axiom A flow (restricted to a nontrivial basic set) then (T ) e hT hT ; as T ! +1 i.e. lim T!+1 (T ) e hT =hT = 1. This generalized a result of Margulis for geodesic flows over manifolds of negative sectional curvature [6]. It is an interesting problem to estimate the error terms in such asymptotic formulae. In the particular case of geodesic flows over compact negatively curved manifolds we showed that there was an exponential error term (with a suitable principal term) [10]. Our first result gives an error term in the case of weakmixing transitive Anosov flows, in which case = M . Theorem 1. Let OE t : M ! M be a weakmixing transitive Anosov flow. Then there exists ffi ? 0 such that (T ) = e hT hT ` 1 + O ` 1 T ffi " : There are examples of Axiom A flows for which the error term may be arbitrarily bad [9]. In particular, it need not be the case that an error term as in the statement of Theorem 1 is satisfied. In order to obtain a positive result, we shall consider flows satisfying the following condition. The authors would like to thank Dmitri Dolgopyat and Francois Ledrappier for useful discussions. Typeset by A M ST E X 1 2 MARK POLLICOTT AND RICHARD SHARP Approximability condition. The flow OE has three closed orbits fl 1 , fl 2 and fl 3 with distinct least periods l(fl 1 ), l(fl 2 ) and l(fl 3 ), respectively, such tha...
The GelfondSchnirelman Method In Prime Number Theory
 Canad. J. Math
"... The original GelfondSchnirelman method, proposed in 1936, uses polynomials with integer coe#cients and small norms on [0, 1] to give a Chebyshevtype lower bound in prime number theory. We study a generalization of this method for polynomials in many variables. Our main result is a lower bound for t ..."
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Cited by 4 (4 self)
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The original GelfondSchnirelman method, proposed in 1936, uses polynomials with integer coe#cients and small norms on [0, 1] to give a Chebyshevtype lower bound in prime number theory. We study a generalization of this method for polynomials in many variables. Our main result is a lower bound for the integral of Chebyshev's #function, expressed in terms of the weighted capacity. This extends previous work of Nair and Chudnovsky, and connects the subject to the potential theory with external fields generated by polynomialtype weights. We also solve the corresponding potential theoretic problem, by finding the extremal measure and its support. 1. Lower bounds for arithmetic functions Let #(x) be the number of primes not exceeding x. The celebrated Prime Number Theorem (PNT), suggested by Legendre and Gauss, states that ##.