相关论文: The Path to Recent Progress on Small Gaps Between …
An overview of the results of new exhaustive computations of gaps between primes in arithmetic progressions is presented. We also give new numerical results for exceptionally large least primes in arithmetic progressions.
We discuss recent advances on weak forms of the Prime $k$-tuple Conjecture, and its role in proving new estimates for the existence of small gaps between primes and the existence of large gaps between primes.
We prove that a positive proportion of the gaps between consecutive primes are short gaps of length less than any fixed fraction of the average spacing between primes.
This is an expository article on the recent marvellous theorem of Goldston, Pintz, and Yildirim on small gaps between prime numbers.
We implement the Maynard-Tao method of detecting primes in tuples to investigate small gaps between primes in arithmetic progressions, with bounds that are uniform over a range of moduli.
The article focuses on the problems of prime gaps and zero spacings. Possible solutions of several related problems such as the greatest lower bound, the least upper bound of the zero spacings, and the least upper bound of the prime gaps…
One of the themes of this paper is recent results on large gaps between primes. The first of these results has been achieved in the paper [12] by Ford, Green, Konyagin and Tao. It was later improved in the joint paper [13] of these four…
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…
In this paper, we study the gaps between primes in Beatty sequences following the methods in the recent breakthrough of J. Maynard.
We show that a positive proportion of all gaps between consecutive primes are small gaps. We provide several quantitative results, some unconditional and some conditional, in this flavour.
These are notes on Zhang's work and subsequent developments produced in preparation for 5 hours of talks for a general mathematical audience given in Cambridge, Edinburgh and Auckland over the last year. Being for colloquium-style talks,…
Most prime gaps results have been proven using tools from analytic or algebraic number theory in the last few centuries. In this paper, we would like to present some probabilistic way of proving many essential results. A major component of…
We show that the existence of arithmetic progressions with few primes, with a quantitative bound on "few", implies the existence of larger gaps between primes less than x than is currently known unconditionally. In particular, we derive…
We consider the problem of finding small prime gaps in various sets of integers $\mathcal{C}$. Following the work of Goldston-Pintz-Yildirim, we will consider collections of natural numbers that are well-controlled in arithmetic…
In this paper we study the integrals of fractional parts of given functions, and develop some new tools to understand the behaviour of prime differences. We demonstrate how simply some seemingly difficult conjectures related to prime…
In a recent joint work with D.A. Goldston and C.Y. Yildirim we just missed by a hairbreadth a proof that bounded gaps between primes occur infinitely often. In the present work it is shown that adding to the primes a much thinner set,…
This is a brief survey of recent results related to austere submanifolds, mainly based on the papers [24,25].
This is an introductory article to the theory of multiple gaps.
In this paper, a survey about recent progress on problems solved using graph amalgamations is presented, along with some new results with complete proofs, and some related open problems.
We show that there exists pairs of consecutive primes less than $x$ whose difference is larger than $t(1+o(1))(\log{x})(\log\log{x})(\log\log\log\log{x})(\log\log\log{x})^{-2}$ for any fixed $t$. Our proof works by incorporating recent…