The Difference Between Consecutive Primes, II

  title={The Difference Between Consecutive Primes, II},
  author={Roger C. Baker and Glyn Harman and Janos Pintz},
  journal={Proceedings of the London Mathematical Society},
The authors sharpen a result of Baker and Harman (1995), showing that [x, x + x0.525] contains prime numbers for large x. An important step in the proof is the application of a theorem of Watt (1995) on a mean value containing the fourth power of the zeta function. 2000 Mathematical Subject Classification: 11N05. 

Short Effective Intervals Containing Primes and a property of the Riemann Zeta Function ζ

In this paper, we prove the existence of primes in the interval ]�, � + 2 �] by inducing an inequality which defines the lower bound of number of primes in the interval ]�, � + 2 �] and suggest an

Arithmetic Properties of Blocks of Consecutive Integers

This paper provides a survey of results on the greatest prime factor, the number of distinct prime factors, the greatest squarefree factor and the greatest m-th powerfree part of a block of

Gaussian primes in Narrow sectors

The purpose of this paper is to show how a sieve method which has had many applications to problems involving rational primes can be modified to derive new results on Gaussian primes (or, more

On Proving of Diophantine Inequalities with Prime Numbers by Evaluations of the Difference between Consecutive Primes

Using as the working hypothesis of an evaluation of the difference between primes $p_{n+1} - p_n = O(\sqrt{p_n})$ we represent in detail the proofs of Legendre's and Oppermann's conjectures.


Let c > 0.55. Every large n can be written in the form p +ab, where p is prime, a and b are significantly smaller than x^1/2 and ab is less than n^c. This strengthens a result of Heath-Brown, which

Primes between consecutive squares

Abstract. A well known conjecture about the distribution of primes asserts that between two consecutive squares there is always at least one prime number. The proof of this conjecture is quite out of

Squarefree Integers And Extreme Values Of Some Arithmetic Functions

A study of the Dedekind psi function concludes that its extreme values are supported on the subset of primorial integers N_k = 2*3***p_k, where p_k is the kth prime. In particular, the inequality

A Structure Theorem for Positive Density Sets Having the Minimal Number of 3-term Arithmetic Progressions

Assuming the well-known conjecture that [x,x+x^t] contains a prime for t > 0 and x sufficiently large, we prove: For 0 0 as s -> 0. A curious feature of the proof is that Behrend's construction on

Diophantine Inequalities as a Problem of Difference between Consecutive Primes

In the present paper, we have developed a method for solving \textit{diophantine inequalities} using their relationship with the \textit{difference between consecutive primes}. Using this approach

Goldbach Numbers in Short Intervals -- A Nonnegative Model Approach

We decrease the length of the shortest interval for which almost all even integers in it are the sum of two primes. This is achieved by applying a version of the Circle Method that uses two minorants



The number of primes in a short interval.

which estimates the number of primes in the interval (x —y, x]. According to the Prime Number Theorem, the above estimate holds uniformly for cx^y^x, if c is any positive constant. Much work has been

Primes of the Form [nc]

Abstract A lower bound for πc(x), the number of primes of the form [nc] in [1, x], is given: πc(x) ⪢ x/log x. The significant point here is the longer range of c than formerly, namely 1 20 17 . The

On the Distribution of ap Modulo One

Abstract In this paper, we prove: "Suppose α is an irrational number and let || y || denote the smallest distance of y from an integer. Then, for any real number β, there are infinitely many primes p

Kloosterman Sums and a Mean Value for Dirichlet Polynomials

Abstract Let a 1 , a 2 , a 3 ,... be complex numbers. Then, for ϵ > 0, M ≥ 1 and T ≥ 1, [formula] where ζ( s ) is Riemann′s zeta-function and C ϵ depends only on ϵ. The proof is based on a technical

On the Exceptional Set for Goldbach's Problem in Short Intervals

Assume the Generalized Riemann Hypothesis and suppose thatHlog−6x→t8. Then we prove that all even integers in any interval of the form (x, x, +H) butO(H1/2log3x) exceptions are a sum of two primes.

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

On the distribution of αp modulo one (II)

  • C. Jia
  • Mathematics, Philosophy
  • 2000
Suppose that α is an irrational number and β is a real number. It is proved that there are infinitely many prime numbers p such that ‖ αp-β‖

Prime Numbers in Short Intervals and a Generalized Vaughan Identity

1. Introduction. Many problems involving prime numbers depend on estimating sums of the form ΣΛ(n)f(n), for appropriate functions f(n), (here, as usual, Λ(n) is the von Mangoldt function). Three

Pintz, `The exceptional set for Goldbach's problem in short intervals', Sieve methods, exponential sums and their applications in number theory (ed

  • 1997

`Primzahlprobleme in der Analysis

  • Sitz. Preuss. Akad. Wiss
  • 1930