Small Gaps between Primes Exist

In the recent preprint [3], Goldston, Pintz, and Y{\i}ld{\i}r{\i}m established, among other things, $$\liminf_{n\to\infty}{p_{n+1}-p_n\over\log p_n}=0,\leqno(0)$$ with $p_n$ the $n$th prime. In the present article, which is essentially self-contained, we shall develop a simplified account of the method used in [3]. While [3] also includes quantitative versions of $(0)$, we are concerned here solely with proving the qualitative $(0)$, which still exhibits all the essentials of the method. We also show here that an improvement of the Bombieri--Vinogradov prime number theorem would give rise infinitely often to bounded differences between consecutive primes. We include a short expository last section. Detailed discussions of quantitative results and a historical review will appear in the publication version of [3] and its continuations.