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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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“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,
An asymptotic formula for the number of smooth values of a polynomial
 J. Number Theory
, 1999
"... Integers without large prime factors, dubbed smooth numbers, are by now firmly established as a useful and versatile tool in number theory. More than being simply a property of numbers that is conceptually dual to primality, smoothness has played a major role in the proofs of many results, from mult ..."
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Integers without large prime factors, dubbed smooth numbers, are by now firmly established as a useful and versatile tool in number theory. More than being simply a property of numbers that is conceptually dual to primality, smoothness has played a major role in the proofs of many results, from multiplicative questions to Waring’s problem to complexity
200?), Small gaps between products of two primes
 arXiv.math.NT/0609615. GAPS BETWEEN ALMOST PRIMES 23
"... As an approximation to the twin prime conjecture it was proved in [11] that (1.1) liminf n→∞ pn+1 − pn ..."
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As an approximation to the twin prime conjecture it was proved in [11] that (1.1) liminf n→∞ pn+1 − pn
An uncertainty principle for arithmetic sequences, preprint, available from www.arxiv.org
"... Abstract. Analytic number theorists usually seek to show that sequences which appear naturally in arithmetic are “welldistributed ” in some appropriate sense. In various discrepancy problems, combinatorics researchers have analyzed limitations to equidistribution, as have Fourier analysts when wor ..."
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Abstract. Analytic number theorists usually seek to show that sequences which appear naturally in arithmetic are “welldistributed ” in some appropriate sense. In various discrepancy problems, combinatorics researchers have analyzed limitations to equidistribution, as have Fourier analysts when working with the “uncertainty principle”. In this article we find that these ideas have a natural setting in the analysis of distributions of sequences in analytic number theory, formulating a general principle, and giving several examples. 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 of the ..."
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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 ...
Rabinowitsch Revisited
"... this paper we are primarily interested in further developing the theory of quadratic polynomials for which many of the small values are prime (rather than "all" as in Rabinowitsch's result). It is wellknown (see [5]) that if the class number of some imaginary quadratic field with large discriminant ..."
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this paper we are primarily interested in further developing the theory of quadratic polynomials for which many of the small values are prime (rather than "all" as in Rabinowitsch's result). It is wellknown (see [5]) that if the class number of some imaginary quadratic field with large discriminant is one then we will have an egregious counterexample to the Generalized Riemann Hypothesis (that is, a zero of the associated Dirichlet Lfunction which is very close to 1). Thus Rabinowitsch's result can be informally stated as "n
THE DISTRIBUTION OF PRIME NUMBERS
, 2006
"... What follows is an expanded version of my lectures at the NATO School on Equidistribution. I have tried to keep the informal style of the lectures. In particular, I have sometimes oversimplified matters in order to convey the spirit of an argument. Lecture 1: The Cramér model and gaps between consec ..."
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What follows is an expanded version of my lectures at the NATO School on Equidistribution. I have tried to keep the informal style of the lectures. In particular, I have sometimes oversimplified matters in order to convey the spirit of an argument. Lecture 1: The Cramér model and gaps between consecutive primes The prime number theorem tells us that π(x), the number of primes below x, is ∼ x / logx. Equivalently, if pn denotes the nth smallest prime number then pn ∼ n log n. What is the distribution of the gaps between consecutive primes, pn+1 − pn? We have just seen that pn+1 − pn is approximately log n “on average”. How often do we get a gap of size 2 logn, say; or of size 1 log n? One way to make this question precise 2 is to fix an interval [α, β] (with 0 ≤ α < β) and ask for
An uncertainty principle for function fields, submitted
"... Abstract. In a recent paper, Granville and Soundararajan [8] proved an “uncertainty principle” for arithmetic sequences, which limits the extent to which such sequences can be welldistributed in both short intervals and arithmetic progressions. In the present paper we follow the methods of [8] and ..."
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Abstract. In a recent paper, Granville and Soundararajan [8] proved an “uncertainty principle” for arithmetic sequences, which limits the extent to which such sequences can be welldistributed in both short intervals and arithmetic progressions. In the present paper we follow the methods of [8] and prove that a similar phenomenon holds in Fq[t]. 1.
Limitations to the Equidistribution of Primes III
 Comp. Math
, 1992
"... : In an earlier paper [FG] we showed that the expected asymptotic formula ß(x; q; a) ¸ ß(x)=OE(q) does not hold uniformly in the range q ! x= log N x, for any fixed N ? 0. There are several reasons to suspect that the expected asymptotic formula might hold, for large values of q, when a is kept f ..."
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: In an earlier paper [FG] we showed that the expected asymptotic formula ß(x; q; a) ¸ ß(x)=OE(q) does not hold uniformly in the range q ! x= log N x, for any fixed N ? 0. There are several reasons to suspect that the expected asymptotic formula might hold, for large values of q, when a is kept fixed. However, by a new construction, we show herein that this fails in the same ranges, for a fixed and, indeed, for almost all a satisfying 0 ! jaj ! x= log N x. 1. Introduction. For any positive integer q and integer a coprime to q, we have the asymptotic formula (1:1) ß(x; q; a) ¸ ß(x) OE(q) as x ! 1, for the number ß(x; q; a) of primes p x with p j a (mod q), where ß(x) is the number of primes x, and OE is Euler's function. In fact (1.1) is known to hold uniformly for (1:2) q ! log N x and all (a; q) = 1, for every fixed N ? 0 (the SiegelWalfisz Theorem), for almost all q ! x 1=2 = log 2+" x and all (a; q) = 1 (the BombieriVinogradov Theorem) and for almost all q !...