• Corpus ID: 118749198

Small Gaps Between Primes I

  title={Small Gaps Between Primes I},
  author={D. A. Goldston and Cem Yalçın Yıldırım},
  journal={arXiv: Number Theory},
We use short divisor sums to approximate prime tuples and moments for primes in short intervals. By connecting these results to classical moment problems we are able to prove that a positive proportion of consecutive primes are within a quarter of the average spacing between primes. 
Arithmetic progressions and the primes
We describe some of the machinery behind recent progress in establishing infinitely many arithmetic progressions of length k in various sets of integers, in particular in arbitrary dense subsets of
Small gaps between the set of products of at most two primes
  • Keiju Sono
  • Mathematics
    Journal of the Mathematical Society of Japan
  • 2020
In this paper, we apply the method of Maynard and Tao to the set of products of two distinct primes (E2-numbers). We obtain several results on the distribution of E2-numbers and primes. Among others,
The prime ideals in every class contain arbitrary large truncated classes
We show that the prime divisors in every class on a projective curve over a finite field contain arbitrary large truncated generalized classes of finite effective divisors.
Prime numbers and L-functions
The classical memoir by Riemann on the zeta function was motivated by questions about the distribution of prime numbers. But there are important problems concerning prime numbers which cannot be
Szemerédi's theorem and problems on arithmetic progressions
Szemeredi's famous theorem on arithmetic progressions asserts that every subset of integers of positive asymptotic density contains arithmetic progressions of arbitrary length. His remarkable theorem
Obstructions to uniformity, and arithmetic patterns in the primes
  • T. Tao
  • Mathematics, Computer Science
  • 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 that the primes contain
The primes contain arbitrarily long polynomial progressions
We establish the existence of infinitely many polynomial progressions in the primes; more precisely, given any integer-valued polynomials P1, …, Pk ∈ Z[m] in one unknown m with P1(0) = … = Pk(0) = 0,
On Rough and Smooth Neighbors
We study the behavior of the arithmetic functions defined by F(n) = P (n) P−(n + 1) and G(n) = P (n + 1) P−(n) (n ≥ 1), where P(k) and P−(k) denote the largest and the smallest prime factors,
The dichotomy between structure and randomness, arithmetic progressions, and the primes
A famous theorem of Szemeredi asserts that all subsets of the integers with positive upper density will contain arbitrarily long arithmetic progressions. There are many different proofs of this deep
Generalising the Hardy-Littlewood Method for Primes
The Hardy�Littlewood method is a well-known technique in analytic number theory. Among its spectacular applications are Vinogradov�s 1937 result that every sufficiently large odd number is a sum of


Higher correlations of divisor sums related to primes I: triple correlations
We obtain the triple correlations for a truncated divisor sum related to primes. We also obtain the mixed correlations for this divisor sum when it is summed over the primes, and give some
Small differences between prime numbers
  • E. Bombieri, H. Davenport
  • Mathematics
    Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences
  • 1966
Let pn denote the nth prime number. The present investigation relates to the existence of relatively small values of pn+1─ pn when n is large, and establishes more precise results than those
On the distribution of primes in short intervals
One of the formulations of the prime number theorem is the statement that the number of primes in an interval ( n , n + h ], averaged over n ≤ N , tends to the limit λ, when N and h tend to infinity
Primes in arithmetic progressions to large moduli
Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
The difference between consecutive prime numbers. II
In a previous paper, under the same title, I considered the problem of how far apart two consecutive primes can be. The present paper is concerned with the opposite question. How near together can
On Bombieri and Davenport's theorem concerning small gaps between primes
§1. Introduction . In this paper we give a new proof of a theorem of Bombieri and Davenport [2, Theorem 1]. Let t (– k ) = t ( k ) be real, where e ( u ) = e 2 πiu . Let p and p ' denote primes, k an
The difference of consecutive primes
is greater than (c2/2)n. A simple calculation now shows that the primes satisfying (4) also satisfy the first inequality of (3) i΀ = e(ci) is chosen small enough. The second inequality of (3) is
The Difference between Consecutive Prime Numbers, III
I = lim inf v n-a* log n The purpose of this paper is to combine the methods used in two earlier papers' in order to prove the following theorem. THEOREM. (1) 1 5 c(1 + 4e)/5, where c 1 -c, so that
The Theory of the Riemann Zeta-Function
The Riemann zeta-function embodies both additive and multiplicative structures in a single function, making it our most important tool in the study of prime numbers. This volume studies all aspects
Some new asymptotic properties for the zeros of Jacobi, Laguerre, and Hermite polynomials
AbstractFor the generalized Jacobi, Laguerre, and Hermite polynomials $$P_n^{\left( {\alpha _n ,\beta _n } \right)} \left( x \right),L_n^{\left( {\alpha _n } \right)} \left( x \right),H_n^{\left(