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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 42 (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,
Open Problems in Number Theoretic Complexity, II
"... this paper contains a list of 36 open problems in numbertheoretic complexity. We expect that none of these problems are easy; we are sure that many of them are hard. This list of problems reflects our own interests and should not be viewed as definitive. As the field changes and becomes deeper, new ..."
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Cited by 30 (0 self)
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this paper contains a list of 36 open problems in numbertheoretic complexity. We expect that none of these problems are easy; we are sure that many of them are hard. This list of problems reflects our own interests and should not be viewed as definitive. As the field changes and becomes deeper, new problems will emerge and old problems will lose favor. Ideally there will be other `open problems' papers in future ANTS proceedings to help guide the field. It is likely that some of the problems presented here will remain open for the forseeable future. However, it is possible in some cases to make progress by solving subproblems, or by establishing reductions between problems, or by settling problems under the assumption of one or more well known hypotheses (e.g. the various extended Riemann hypotheses, NP 6= P; NP 6= coNP). For the sake of clarity we have often chosen to state a specific version of a problem rather than a general one. For example, questions about the integers modulo a prime often have natural generalizations to arbitrary finite fields, to arbitrary cyclic groups, or to problems with a composite modulus. Questions about the integers often have natural generalizations to the ring of integers in an algebraic number field, and questions about elliptic curves often generalize to arbitrary curves or abelian varieties. The problems presented here arose from many different places and times. To those whose research has generated these problems or has contributed to our present understanding of them but to whom inadequate acknowledgement is given here, we apologize. Our list of open problems is derived from an earlier `open problems' paper we wrote in 1986 [AM86]. When we wrote the first version of this paper, we feared that the problems presented were so difficult...
The distribution of values of L(1, χd
 Geom. Funct. Anal
"... Throughout this paper d will denote a fundamental discriminant, and χd the associated primitive real character to the modulus d. We investigate here the distribution of values of L(1, χd) as d varies over all fundamental discriminants with d  ≤ x. Our main concern is to compare the distribution ..."
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Cited by 26 (4 self)
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Throughout this paper d will denote a fundamental discriminant, and χd the associated primitive real character to the modulus d. We investigate here the distribution of values of L(1, χd) as d varies over all fundamental discriminants with d  ≤ x. Our main concern is to compare the distribution of values of L(1, χd) with the distribution of “random Euler
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 25 (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.
Explicit Bounds on Exponential Sums and the Scarcity of Squarefree Binomial Coefficients
, 1996
"... This paper fills what we believe to be a lacuna in the existing literature concerning upper bounds on exponential sums. Although it has always been evident that many of the known estimates can be made explicit, it is a nontrivial problem to actually do so. In particular so that the constants involv ..."
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Cited by 25 (1 self)
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This paper fills what we believe to be a lacuna in the existing literature concerning upper bounds on exponential sums. Although it has always been evident that many of the known estimates can be made explicit, it is a nontrivial problem to actually do so. In particular so that the constants involved do not render the explicit estimates useless in practical applications. We have used the practical bounds that are needed to prove Theorem 1 as motivation for our results here, though we hope that this work will be applicable to a variety of other problems which routinely apply these or related exponential sum estimates. In particular our results here can be used to say something about the questions of estimating the number of integers free of large prime factors in short intervals (see [FL]), and of the largest prime factor of an integer in an interval (see [J]). Our key result is
It’s as easy as ABC
 Notices of the AMS
"... In this age in which mathematicians are supposed to bring their research into the classroom, even at the most elementary level, it is rare that we can turn the tables and use our elementary teaching to help in our research. However, in giving a proof of Fermat’s Last Theorem, it turns out that we ca ..."
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Cited by 24 (2 self)
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In this age in which mathematicians are supposed to bring their research into the classroom, even at the most elementary level, it is rare that we can turn the tables and use our elementary teaching to help in our research. However, in giving a proof of Fermat’s Last Theorem, it turns out that we can use tools from calculus and linear algebra only. This may strike some readers as unlikely, but bear with us for a few moments as we give our proof. Fermat claimed that there are no solutions to (1) x p + y p = z p for p ≥ 3, with x, y, and z all nonzero. If we assume that there are solutions to (1), then we can assume that x, y, and z have no common factor, else we can divide out by that factor. Our first step will be to differentiate (1) to get px p−1 x ′ + py p−1 y ′ = pz p−1 z ′, and after dividing out the common factor p, this leaves us with
Small gaps between prime numbers: the work of GoldstonPintzYıldırım
, 2000
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