• Corpus ID: 119174212

On the difference between consecutive primes

  title={On the difference between consecutive primes},
  author={James Maynard},
  journal={arXiv: Number Theory},
  • J. Maynard
  • Published 9 January 2012
  • Mathematics
  • arXiv: Number Theory
Update: This work reproduces an earlier result of Peck, which the author was initially unaware of. The method of the proof is essentially the same as the original work of Peck. There are no new results. We show that the sum of squares of differences between consecutive primes $\sum_{p_n\le x}(p_{n+1}-p_n)^2$ is bounded by $x^{5/4+{\epsilon}}$ for $x$ sufficiently large and any fixed ${\epsilon}>0$. The proof relies on utilising various mean-value estimates for Dirichlet polynomials. 

An observation on the difference between consecutive primes

Consider two consecutive odd primes $p_n$ and $p_{n+1}$, let $m$ to be their midpoint, fixed once for all. We prove unconditionally that every $x$ in the interval $[\frac{\ln{(m-p_n)}}{\log{p_n}},

Some heuristics on the gaps between consecutive primes

We propose the formula for the number of pairs of consecutive primes $p_n, p_{n+1}<x$ separated by gap $d=p_{n+1}-p_n$ expressed directly by the number of all primes $<x$, i.e. by $\pi(x)$. As the

An Explicit Result for Primes Between Cubes

We prove that there is a prime between $n^3$ and $(n+1)^3$ for all $n \geq \exp(\exp(33.217))$. Our new tool which we derive is a version of Landau's explicit formula for the Riemann zeta-function

The Differences Between Consecutive Primes. V

We show that \[\sum_{\substack{p_n\le x\\ p_{n+1}-p_n\ge\sqrt{p_n}}}(p_{n+1}-p_n)\ll_{\varepsilon} x^{3/5+\varepsilon}\] for any fixed $\varepsilon>0$. This improves a result of Matomaki, in which

A Short Note on Gaps between Powers of Consecutive Primes

Let $\alpha, \beta \geq 0$ and $\alpha + \beta < 1$. In this short note, we show that $\liminf_{n \to \infty} p_n^\beta(p_{n+1}^\alpha - p_n^\alpha) = 0$, where $p_n$ is the $n$th prime. This notes

Explicit Estimates in the Theory of Prime Numbers

It is the purpose of this thesis to enunciate and prove a collection of explicit results in the theory of prime numbers. First, the problem of primes in short intervals is considered. We prove that

Conditional results about primes between consecutive powers

A well known conjecture about the distribution of primes asserts that all intervals of type [n, (n+1)] contain at least one prime. The proof of this conjecture is quite out of reach at present, even

On sums of powers of almost equal primes

On the Möbius function in all short intervals

We show that, for the Mobius function $\mu(n)$, we have $$ \sum_{x 0.55$. This improves on a result of Ramachandra from 1976, which is valid for $\theta>7/12$. Ramachandra's result corresponded to

Averaged Form of the Hardy-Littlewood Conjecture

We study the prime pair counting functions $\pi_{2k}(x),$ and their averages over $2k.$ We show that good results can be achieved with relatively little effort by considering averages. We prove an



On the difference between consecutive primes

[Received 3 August 1937] 1. LET pn denote the nth. prime, and 77(2;) the number of primes p not exceeding x. The existence of an absolute constant 6 < 1 such that a (1) n(x) ~ . log a; when x -> 00,

The Differences Between Consecutive Primes

This paper discusses two problems which relate to the difference between consecu~ve primes. Let p. be the nth prime, and d. = p . + l p . . (1) We want to find a funct ion f(,~) such that d . 0 is a

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 V

  • R. Rankin
  • Mathematics
    Proceedings of the Edinburgh Mathematical Society
  • 1963
Let pn denote the nth prime and let ε be any positive number. In 1938 (3) Ishowed that, for an infinity of values of n, where, for k≧1, logk+1x = log (logk x) and log1x = log x. In a recent paper (4)

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

On the difference between consecutive prime numbers

/ = lim inf ̂ 11-tl . »->» log pn The purpose of this paper is to combine the methods used in two earlier papers1 in order to prove the following theorem. Theorem. (1) / = c(l + 40)/5, where c<

Very Large Gaps between Consecutive Primes

Abstract LetG(X) denote the largest gap between consecutive primes belowX. Improving earlier results of Erdős, Rankin, Schonhage, and Maier-Pomerance, we prove G(X)⩾(2e γ +o(1)) log Xlog 2 Xlog 4

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

On maximal gaps between successive primes

exists and perhaps it might be possible to determine its value, but it will probably not be possible to express ft(n) by a simple function of n and t (even for t = 3). If t is large compared to n our


er. We shall see how their legacy has inuenced research for most of the rest of the century, particularly through the 'schools' of Selberg, and of Erdos, and with the \large sieve" in the sixties.