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 \(\Delta = 0\).