Results 1  10
of
21
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 ..."
Abstract

Cited by 41 (2 self)
 Add to MetaCart
“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,
The GreenTao Theorem on arithmetic progressions in the primes: an ergodic point of view
, 2005
"... A longstanding and almost folkloric conjecture is that the primes contain arbitrarily long arithmetic progressions. Until recently, the only progress on this conjecture was due to van der Corput, who showed in 1939 that there are infinitely many triples of primes in arithmetic progression. In an a ..."
Abstract

Cited by 31 (2 self)
 Add to MetaCart
(Show Context)
A longstanding and almost folkloric conjecture is that the primes contain arbitrarily long arithmetic progressions. Until recently, the only progress on this conjecture was due to van der Corput, who showed in 1939 that there are infinitely many triples of primes in arithmetic progression. In an amazing fusion of methods from analytic number theory and ergodic theory, Ben Green and Terence Tao showed that for any positive integer k, there exist infinitely many arithmetic progressions of length k consisting only of prime numbers. This is an introduction to some of the ideas in the proof, concentrating on the connections to ergodic theory.
Period of the power generator and small values of Carmichael’s function
 Math.Comp.,70
"... Abstract. Consider the pseudorandom number generator un ≡ u e n−1 (mod m), 0 ≤ un ≤ m − 1, n =1, 2,..., where we are given the modulus m, the initial value u0 = ϑ and the exponent e. One case of particular interest is when the modulus m is of the form pl, where p, l are different primes of the same ..."
Abstract

Cited by 26 (12 self)
 Add to MetaCart
(Show Context)
Abstract. Consider the pseudorandom number generator un ≡ u e n−1 (mod m), 0 ≤ un ≤ m − 1, n =1, 2,..., where we are given the modulus m, the initial value u0 = ϑ and the exponent e. One case of particular interest is when the modulus m is of the form pl, where p, l are different primes of the same magnitude. It is known from work of the first and third authors that for moduli m = pl, if the period of the sequence (un) exceeds m3/4+ε, then the sequence is uniformly distributed. We show rigorously that for almost all choices of p, l it is the case that for almost all choices of ϑ, e, the period of the power generator exceeds (pl) 1−ε. And so, in this case, the power generator is uniformly distributed. We also give some other cryptographic applications, namely, to rulingout the cycling attack on the RSA cryptosystem and to socalled timerelease crypto. The principal tool is an estimate related to the Carmichael function λ(m), the size of the largest cyclic subgroup of the multiplicative group of residues modulo m. In particular, we show that for any ∆ ≥ (log log N) 3,wehave λ(m) ≥ N exp(−∆) for all integers m with 1 ≤ m ≤ N, apartfromatmost N exp −0.69 ( ∆ log ∆) 1/3) exceptions. 1.
Average twin prime conjecture for elliptic curves
, 2007
"... Let E be an elliptic curve over Q. In 1988, Koblitz conjectured a precise asymptotic for the number of primes p up to x such that the order of the group of points of E over Fp is prime. This is an analogue of the Hardy and Littlewood twin prime conjecture in the case of elliptic curves. Koblitz’s co ..."
Abstract

Cited by 21 (6 self)
 Add to MetaCart
(Show Context)
Let E be an elliptic curve over Q. In 1988, Koblitz conjectured a precise asymptotic for the number of primes p up to x such that the order of the group of points of E over Fp is prime. This is an analogue of the Hardy and Littlewood twin prime conjecture in the case of elliptic curves. Koblitz’s conjecture is still widely open. In this paper we prove that Koblitz’s conjecture is true on average over a twoparameter family of elliptic curves. One of the key ingredients in the proof is a short average distribution result in the style of BarbanDavenportHalberstam,
Obstructions to uniformity, and arithmetic patterns in the primes
, 2005
"... In this expository article, we describe the recent approach, motivated by ergodic theory, towards detecting arithmetic patterns in the primes, and in particular establishing in [26] that the primes contain arbitrarily long arithmetic progressions. One of the driving philosophies is to identify prec ..."
Abstract

Cited by 9 (5 self)
 Add to MetaCart
In this expository article, we describe the recent approach, motivated by ergodic theory, towards detecting arithmetic patterns in the primes, and in particular establishing in [26] that the primes contain arbitrarily long arithmetic progressions. One of the driving philosophies is to identify precisely what the obstructions could be that prevent the primes (or any other set) from behaving “randomly”, and then either show that the obstructions do not actually occur, or else convert the obstructions into usable structural information on the primes.
Avoiding monochromatic sequences with special gaps, preprint
"... Abstract. For S ⊆ Z + and k and r fixed positive integers, denote by f(S, k; r) the least positive integer n (if it exists) such that within every rcoloring of {1, 2,..., n} there must be a monochromatic sequence {x1, x2,..., xk} with xi − xi−1 ∈ S for 2 ≤ i ≤ k. We consider the existence of f(S, k ..."
Abstract

Cited by 5 (3 self)
 Add to MetaCart
(Show Context)
Abstract. For S ⊆ Z + and k and r fixed positive integers, denote by f(S, k; r) the least positive integer n (if it exists) such that within every rcoloring of {1, 2,..., n} there must be a monochromatic sequence {x1, x2,..., xk} with xi − xi−1 ∈ S for 2 ≤ i ≤ k. We consider the existence of f(S, k; r) for various choices of S, as well as upper and lower bounds on this function. In particular, we show that this function exists for all k if S is an odd translate of the set of primes and r = 2.
Prime number patterns
 Amer. Math. Monthly
"... and that there are infinitely many primes in any arithmetic progression a, a + d, a + 2d,... provided gcd(a, d) = 1andd ≥ 1. If we ask slightly more involved questions, such as whether there exist infinitely many primes of the form n 2 + 1, or infinitely many pairs of primes of the form p, p + 2, t ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
(Show Context)
and that there are infinitely many primes in any arithmetic progression a, a + d, a + 2d,... provided gcd(a, d) = 1andd ≥ 1. If we ask slightly more involved questions, such as whether there exist infinitely many primes of the form n 2 + 1, or infinitely many pairs of primes of the form p, p + 2, then these questions are open,
On the ternary Goldbach problem with primes in independent arithmetic progressions
 Acta Math. Hungar
"... For A,ε> 0 and any sufficiently large odd n we show that for almost all k ≤ R: = n 1/5−ε there exists a representation n = p1 + p2 + p3 with primes pi ≡ bi mod k for almost all admissible triplets b1,b2,b3 of reduced residues mod k. ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
(Show Context)
For A,ε> 0 and any sufficiently large odd n we show that for almost all k ≤ R: = n 1/5−ε there exists a representation n = p1 + p2 + p3 with primes pi ≡ bi mod k for almost all admissible triplets b1,b2,b3 of reduced residues mod k.