Small Gaps between Primes Exist

In the preprint [3], Goldston, Pintz, and Yıldırım established, among other things, (0) lim inf n→∞ pn+1 − pn log pn = 0, with pn the nth prime. In the present article, which is essentially self-contained, we shall develop a simplified account of the method used in [3]. We include a short expository last section. Key word: Prime number. 1. Basic lemma. In this section we shall prove an asymptotic formula relevant to Selberg’s sieve, which is to be modified so as to involve primes in the next section. The two asymptotic formulas thus obtained will be combined in a simple weighted sieve setting, and give rise to (0) in the third section. Let N be a parameter increasing monotonically to infinity. There are four other basic parameters H,R, k, ` in our discussion. We impose the following conditions to them: (1.1) H logN logR ≤ logN, and (1.2) integers k, ` > 0 are arbitrary but bounded. To prove a quantitative assertion superseding (0), we need to regard k, ` as functions of N ; but for our present purpose the above is sufficient (this aspect is to be discussed in the publication version of [3] and its continuations). All implicit constants in the sequel are possibly dependent on k, ` at most; and besides, the symbol c stands for a positive constant with the same dependency, whose value may differ at each occurrence. 2000 Mathematics Subject Classification. Primary 11N05; Secondary 11P32. ∗) Department of Mathematics, San Jose State University, San Jose, CA 95192, USA. ∗∗) Department of Mathematics, College of Science and Technology, Nihon University, Surugadai, Tokyo 101-8308, Japan. ∗∗∗) Rényi Mathematical Institute of the Hungarian Academy of Sciences, H-1364 Budapest, P.O.B. 127, Hungary. ∗∗∗∗) Department of Mathematics, Bog̃aziçi University, Bebek, Istanbul 34342 & Feza Gürsey Enstitüsü, Çengelköy, Istanbul, P.K. 6, 81220, Turkey. Let H = {h1, h2, . . . , hk} ⊆ [1,H] ∩ Z, with hi 6= hj for i 6= j. For a prime p, let Ω(p) be the set of different residue classes among −h (mod p), h ∈ H, and write n ∈ Ω(p) instead of n (mod p) ∈ Ω(p). We call H admissible if (1.3) |Ω(p)| < p for all p, and assume this unless otherwise stated. We extend Ω multiplicatively, so that n ∈ Ω(d) with squarefree d if and only if n ∈ Ω(p) for all p|d, which is equivalent to d|P (n;H) with P (n;H) = (n+h1)(n+ h2) · · · (n + hk). Also, we put, with μ the Möbius function, λR(d; a) = 0 if d > R, 1 a! μ(d) ( log R d )a if d ≤ R,