New equidistribution estimates of Zhang type
In May 2013, Y. Zhang [52] proved the existence of infinitely many pairs of primes with bounded gaps. In particular, he showed that there exists at least one h ě 2 such that the set tp prime | p` h is primeu is infinite. (In fact, he showed this for some even h between 2 and 7ˆ 10, although the precise value of h could not be extracted from his method.) Zhang’s work started from the method of Goldston, Pintz and Yıldırım [23], who had earlier proved the bounded gap property, conditionally on distribution estimates concerning primes in arithmetic progressions to large moduli, i.e., beyond the reach of the Bombieri–Vinogradov theorem. Based on work of Fouvry and Iwaniec [11, 12, 13, 14] and Bombieri, Friedlander and Iwaniec [3, 4, 5], distribution estimates going beyond the Bombieri–Vinogradov range for arithmetic functions such as the von Mangoldt function were already known. However, they involved restrictions concerning the residue classes which were incompatible with the method of Goldston, Pintz and Yıldırım. Zhang’s resolution of this difficulty proceeded in two stages. First, he isolated a weaker distribution estimate that sufficed to obtain the bounded gap property (still