zbMATH — the first resource for mathematics

Small gaps between primes exist. (English) Zbl 1168.11041
Let \(p_n\) denote the \(n\)th prime number. The differences \(p_{n+1} - p_n\) are one of the central objects of study in the theory of distribution of primes. In this paper, the authors are concerned with the limit \[ \Delta = \liminf_{n \to \infty} \frac {p_{n+1} - p_n}{\log p_n}. \] Non-trivial upper bounds for this limit have long been considered approximations to the twin-prime conjecture, and several such bounds have been obtained over the years. Because of the twin-prime conjecture, it was conjectured that \(\Delta = 0\), but until recently even this weaker conjecture was considered well beyond the reach of present methods. That changed in late 2004, when Goldston, Pintz and Yıldırım proved that \(\Delta = 0\). Their original proof, together with proofs of a number of other related – and equally exciting results – will appear in a series of papers entitled Primes in tuples [I, Ann. Math. (2) 170, No. 2, 819–862 (2009; Zbl 1207.11096), II, Acta Math. 204, No. 1, 1–47 (2010; Zbl 1207.11097), III, Funct. Approximatio, Comment. Math. 35, 79–89 (2006; Zbl 1196.11123)].
In the paper under review, the authors give an independent, simplified (and essentially self-contained) proof that \[ \Delta \leq \max\{ 0, 2\theta - 1\}, \eqno{(*)} \] where \(\theta\) is any real number with the following property: Given any fixed \(A > 0\), \[ \sum_{q \leq x^{\theta}} \max_{y \leq x} \max_{a: (a,q)=1} \left| \sum_{_{\substack{ p \leq y\\ p \equiv a \pmod q}}} \log p - \frac y{\phi(q)} \right| \ll \frac x{(\log x)^A}. \] Since the Bombieri-Vinogradov theorem allows us to take \(\theta\) arbitrarily close to \(1/2\), inequality (*) above establishes that \(\Delta = 0\).

11N05 Distribution of primes
11N36 Applications of sieve methods
Full Text: DOI Euclid
[1] E. Bombieri, Le grand crible dans la théorie analytique des nombres , second édition revue et augmentée, Astérisque, 18, Soc. Math. France, Paris, 1987. · Zbl 0618.10042
[2] P. X. Gallagher, On the distribution of primes in short intervals, Mathematika 23 (1976), no. 1, 4-9; Corrigendum, ibid, 28 (1981), 86. · Zbl 0346.10024
[3] D. A. Goldston, J. Pintz, and C. Y. Y\ild\ir\im, Small gaps between primes II (Preliminary). (February 8, 2005). See also [2005-19 of http://aimath.org/preprints.html].
[4] E. C. Titchmarsh, The theory of the Riemann zeta-function , Clarendon Press, Oxford, 1951. · Zbl 0042.07901
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.