Bounded gaps between primes

New equidistribution estimates of Zhang type

“…In Zhang's work [52], the claims Type I r , δ, σs, Type II r , δs,Type III r , δ, σs are (implicitly) proven with " δ " 1{1168 and σ " 1{8´8 . In fact, if one optimizes the numerology in his arguments, one can derive Type I r , δ, σs whenever 44 `12δ`8σ ă 1, Type II r , δs whenever 116 `20δ ă 1, and Type III r , δ, σs whenever σ ą `2 13 δ (see [45] for details).…”

“…Finally, for the Type III sums, Zhang's delicate argument [52] adapts and improves the work of Friedlander and Iwaniec [21] on the ternary divisor function in arithmetic progressions. As we said, it ultimately relies on a three-variable exponential sum estimate that was proved by Birch and Bombieri in the Appendix to [21].…”

“…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 [52], Zhang establishes a weaker form of Theorem 1.1, with θ " 1{2`1{584, and with the a q required to be roots of a polynomial P of the form P pnq :" ś 1ďjďk;j‰i pn`h j´hi q for a fixed admissible tuple ph 1 , . .…”

A note on small gaps between primes in arithmetic progressions

“…Recently, there have been breakthrough developments towards proving this conjecture. First Zhang [8], refining a method of Goldston, Pintz and Yıldırım [3], proved that for k large enough, the sets n+H contain two primes infinitely often, thus settling the bounded gaps conjecture. Then Maynard [5] and Tao (unpublished) independently devised another modification of the Goldston-Pintz-Yıldırım method which could detect m primes in k-tuples for any m, provided k is large enough.…”