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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,
Primes in short intervals
 Commun. Math. Phys
"... Dedicated to Freeman Dyson, with best wishes on the occasion of his eightieth birthday. Abstract. Contrary to what would be predicted on the basis of Cramér’s model concerning the distribution of prime numbers, we develop evidence that the distribution of ψ(x + H) − ψ(x), for 0 ≤ x ≤ N, is approxima ..."
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Dedicated to Freeman Dyson, with best wishes on the occasion of his eightieth birthday. Abstract. Contrary to what would be predicted on the basis of Cramér’s model concerning the distribution of prime numbers, we develop evidence that the distribution of ψ(x + H) − ψ(x), for 0 ≤ x ≤ N, is approximately normal with mean ∼ H and variance ∼ H log N/H, when N δ ≤ H ≤ N 1−δ. Cramér [4] modeled the distribution of prime numbers by independent random variables Xn (for n ≥ 3) that take the value 1 (n is “prime”) with probability 1 / logn and take the value 0 (n is “composite”) with probability 1 − 1 / log n. If pn denotes the n th prime
RSA Key Generation with Verifiable Randomness
 In Public Key Cryptography 2002, LNCS 2274
, 2002
"... Abstract. We consider the problem of proving that a user has selected and correctly employed a truly random seed in the generation of her RSA key pair. This task is related to the problem of key validation, the process whereby a user proves to another party that her key pair has been generated secur ..."
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Abstract. We consider the problem of proving that a user has selected and correctly employed a truly random seed in the generation of her RSA key pair. This task is related to the problem of key validation, the process whereby a user proves to another party that her key pair has been generated securely. The aim of key validation is to pursuade the verifying party that the user has not intentionally weakened or reused her key or unintentionally made use of bad software. Previous approaches to this problem have been ad hoc, aiming to prove that a private key is secure against specific types of attacks, e.g., that an RSA modulus is resistant to ellipticcurvebased factoring attacks. This approach results in a rather unsatisfying laundry list of security tests for keys. We propose a new approach that we refer to as key generation with verifiable randomness (KEGVER). Our aim is to show in zero knowledge that a private key has been generated at random according to a prescribed process, and is therefore likely to benefit from the full strength of the underlying cryptosystem. Our proposal may be viewed as a kind of distributed key generation protocol involving the user and verifying party. Because the resulting private key is held solely by the user, however, we are able to propose a protocol much more practical than conventional distributed key generation. We focus here on a KEGVER protocol for RSA key generation. Key words: certificate authority, key generation, nonrepudiation, publickey infrastructure, verifiable randomness, zero knowledge 1
An uncertainty principle for arithmetic sequences
, 2004
"... 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 ..."
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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.
Irregularities in the distribution of primes in function fields
"... We adapt the Maier matrix method to the polynomial ring Fq[t], and prove analogues of results of Maier [4] and Shiu [10] concerning the distribution of primes in short intervals. ..."
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We adapt the Maier matrix method to the polynomial ring Fq[t], and prove analogues of results of Maier [4] and Shiu [10] concerning the distribution of primes in short intervals.
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
Iterated absolute values of differences of consecutive primes
 Mathematics of Computation
, 1993
"... Abstract. Let dç,(n) = p „ , the nth prime, for n> 1, and let dk+x(n) = \dk(n) dk(n + 1)  for k> 0, n> 1. A wellknown conjecture, usually ascribed to Gilbreath but actually due to Proth in the 19th century, says that dk(\) = 1 for all k> 1. This paper reports on a computation that verified this ..."
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Abstract. Let dç,(n) = p „ , the nth prime, for n> 1, and let dk+x(n) = \dk(n) dk(n + 1)  for k> 0, n> 1. A wellknown conjecture, usually ascribed to Gilbreath but actually due to Proth in the 19th century, says that dk(\) = 1 for all k> 1. This paper reports on a computation that verified this conjecture for k < tt(1013) » 3 x 10 ". It also discusses the evidence and the heuristics about this conjecture. It is very likely that similar conjectures are also valid for many other integer sequences. 1.
Cycle lengths in a permutation are typically Poisson
"... The set of cycle lengths of almost all permutations in Sn are “Poisson distributed”: we show that this remains true even when we restrict the number of cycles in the permutation. The formulas we develop allow us to also show that almost all permutations with a given number of cycles have a certain “ ..."
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The set of cycle lengths of almost all permutations in Sn are “Poisson distributed”: we show that this remains true even when we restrict the number of cycles in the permutation. The formulas we develop allow us to also show that almost all permutations with a given number of cycles have a certain “normal order” (in the spirit of the ErdősTurán theorem). Our results were inspired by analogous questions about the size of the prime divisors of “typical ” integers. 1
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
STRINGS OF CONSECUTIVE PRIMES IN FUNCTION FIELDS
"... Abstract. In a recent paper, Thorne [5] proved the existence of arbitrarily long strings of consecutive primes in arithmetic progressions in the polynomial ring Fq[t]. Here we extend this result to show that given any k there exists a string of k consecutive primes of degree D in arithmetic progress ..."
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Abstract. In a recent paper, Thorne [5] proved the existence of arbitrarily long strings of consecutive primes in arithmetic progressions in the polynomial ring Fq[t]. Here we extend this result to show that given any k there exists a string of k consecutive primes of degree D in arithmetic progression for all sufficiently large D. 1.