Results 1  10
of
20
Almost All Primes Can be Quickly Certified
"... This paper presents a new probabilistic primality test. Upon termination the test outputs "composite" or "prime", along with a short proof of correctness, which can be verified in deterministic polynomial time. The test is different from the tests of Miller [M], SolovayStrassen ..."
Abstract

Cited by 71 (4 self)
 Add to MetaCart
This paper presents a new probabilistic primality test. Upon termination the test outputs "composite" or "prime", along with a short proof of correctness, which can be verified in deterministic polynomial time. The test is different from the tests of Miller [M], SolovayStrassen [SSI, and Rabin [R] in that its assertions of primality are certain, rather than being correct with high probability or dependent on an unproven assumption. Thc test terminates in expected polynomial time on all but at most an exponentially vanishing fraction of the inputs of length k, for every k. This result implies: • There exist an infinite set of primes which can be recognized in expected polynomial time. • Large certified primes can be generated in expected polynomial time. Under a very plausible condition on the distribution of primes in "small" intervals, the proposed algorithm can be shown'to run in expected polynomial time on every input. This
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 20 (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,
Dedekind Zeta Functions and the Complexity of Hilbert’s Nullstellensatz
, 2008
"... Let HN denote the problem of determining whether a system of multivariate polynomials with integer coefficients has a complex root. It has long been known that HN ∈P = ⇒ P =NP and, thanks to recent work of Koiran, it is now known that the truth of the Generalized Riemann Hypothesis (GRH) yields the ..."
Abstract

Cited by 5 (4 self)
 Add to MetaCart
Let HN denote the problem of determining whether a system of multivariate polynomials with integer coefficients has a complex root. It has long been known that HN ∈P = ⇒ P =NP and, thanks to recent work of Koiran, it is now known that the truth of the Generalized Riemann Hypothesis (GRH) yields the implication HN ̸∈P = ⇒ P ̸=NP. We show that the assumption of GRH in the latter implication can be replaced by either of two more plausible hypotheses from analytic number theory. The first is an effective short interval Prime Ideal Theorem with explicit dependence on the underlying field, while the second can be interpreted as a quantitative statement on the higher moments of the zeroes of Dedekind zeta functions. In particular, both assumptions can still hold even if GRH is false. We thus obtain a new application of Dedekind zero estimates to computational algebraic geometry. Along the way, we also apply recent explicit algebraic and analytic estimates, some due to Silberman and Sombra, which may be of independent interest.
A note on the number of primes in short intervals
 Proc. Amer. Math. Soc
, 1990
"... Abstract. Let J(ß,T) = f [ (72x<pi<<x+x/TloZP ~ x/T)2dx/x2, where the sum is over prime powers. H. L. Montgomery has shown that on the Riemann hypothesis, there is a positive constant Co such that for each ß> 1, J{ß,T) < Co/Hog2 T/T, provided that T is sufficiently large. Here we pro ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
Abstract. Let J(ß,T) = f [ (72x<pi<<x+x/TloZP ~ x/T)2dx/x2, where the sum is over prime powers. H. L. Montgomery has shown that on the Riemann hypothesis, there is a positive constant Co such that for each ß> 1, J{ß,T) < Co/Hog2 T/T, provided that T is sufficiently large. Here we prove a slightly stronger result from which we deduce a lower bound of the same order. 1.
Turán’s problem 10 revisited
, 2008
"... In this paper we prove that inf max n∑ z ν k ∣ = √ n + O ( n 0.2625+ǫ). (ǫ> 0) zk≥1 ν=1,...,n2 k=1 This improves on the bound inf max n∑ zk≥1 ν=1,...,n2 z k=1 ν k ∣ ≤ √ 6n log(1 + n2) of Erdős and Renyi. In the special case of n + 1 being a prime we have previously obtained the much sharp ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
In this paper we prove that inf max n∑ z ν k ∣ = √ n + O ( n 0.2625+ǫ). (ǫ> 0) zk≥1 ν=1,...,n2 k=1 This improves on the bound inf max n∑ zk≥1 ν=1,...,n2 z k=1 ν k ∣ ≤ √ 6n log(1 + n2) of Erdős and Renyi. In the special case of n + 1 being a prime we have previously obtained the much sharper result n ≤ inf max n∑ z ν k
On a Class of Relatively Prime Sequences
, 1978
"... For each natural number n, let noOr) = II, and if a,(u),..., a,()$) have already been defined, let ai+l(n)> o,(n) be minimal with (ai+l(n), a,,(n)... a,(n)) = 1. Let g(n) be the largest o,(~I) not a prime oi the square of a prime. We show that g(n), II and that g(n) IJ II + cn’/ * lo&r) fo ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
For each natural number n, let noOr) = II, and if a,(u),..., a,()$) have already been defined, let ai+l(n)> o,(n) be minimal with (ai+l(n), a,,(n)... a,(n)) = 1. Let g(n) be the largest o,(~I) not a prime oi the square of a prime. We show that g(n), II and that g(n) IJ II + cn’/ * lo&r) for some c> 0. The true order of magnitude of g(n) n seems to be connected with the fine distribution of prime numbers. We also show that “most ” U,(II) that are not primes or squares of primes are products of two distinct primes. A result of independent interest comes of one of our droofs: For every sufficiently large n there is a prime p < n”’ with [n/p] composite.
Expander Graphs and Gaps between Primes
"... The explicit construction of infinite families of dregular graphs which are Ramanujan is known only in the case d−1 is a prime power. In this paper, we consider the case when d − 1 is not a prime power. The main result is that by perturbing known Ramanujan graphs and using results about gaps betwee ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
The explicit construction of infinite families of dregular graphs which are Ramanujan is known only in the case d−1 is a prime power. In this paper, we consider the case when d − 1 is not a prime power. The main result is that by perturbing known Ramanujan graphs and using results about gaps between consecutive primes, we are able to construct infinite families of “almost ” Ramanujan graphs for almost every value of d. More precisely, for any fixed ǫ> 0 and for almost every value of d (in the sense of natural density), there are infinitely many dregular graphs such that all the nontrivial eigenvalues of the adjacency matrices of these graphs have absolute value less than (2 + ǫ) √ d − 1. 1