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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 $$\Delta = 0$$.

##### MSC:
 11N05 Distribution of primes 11N36 Applications of sieve methods
##### Keywords:
gaps between primes; sieve methods; prime tuples
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##### References:
 [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
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