Theorem (Zhang, 2013). There exists such that there are infinitely many pairs of primes such that .
Zhang gave . The twin prime conjecture claims that There’s an ongoing Polymath8 project dedicated to lowering , with the currently accepted record being .
Zhang’s idea was to marry two large bodies of work. The first, initiated by Selberg in the 1940s and carried on by Goldston-Pintz-Yildirim in 2005, investigates the link between the distribution of primes in arithmetic progression and bounded gaps. The second, whose numerous contributors included Bombieri, A.I. Vinogradov, Linnik, Heath-Brown, Iwaniec, Friedlander and Fouvry, studies how primes behave in APs.
For primes in AP: we expect that
whenever and . This is only known for , where is any positive number. Roughly speaking GRH is equivalent to (1) holding for . Powerful work of Bombieri and Vinogradov from the 1960s establishes (1) for almost all .
Then in 2005, Goldston, Pintz and Yildirim showed that if (1) holds for almost all (some then . Motohashi and Pintz adapted this to show that it suffices for almost all smooth to satisfy (1); Zhang appears to have done this independently.
In the 1980s, Bombieri, Fouvry, Friendlander and Iwaniec demonstrated (1) for almost all smooth , but this required to be fixed. Zhang removed the technical condition of needing to be fixed.
To tackle the resulting Kloosterman sums BFFI used estimates from automorphic forms, together with Weil bounds, whereas Zhang used only the latter. We expect square root cancellation of these sums, as if primes were distributed randomly, and indeed for prime this is obtained via Weil bounds:
Crucially, if is composite then we can surpass square root cancellation just slightly.
The remainder of the talk elaborates on GPY. Let EH() be the assertion that (1) holds whenever . We briefly recap. The Elliott-Halberstam conjecture is that EH() holds for any . The Bombieri-Vinogradov theorem is that EH() holds whenever . GPY show that if EH() holds for some then . So GPY fell just short of proving bounded prime gaps. GPY also showed, conditionally on EH, that .
A set is admissible if, modulo any prime , the set misses a residue class modulo . This condition is necessary for it to be possible that for infinitely many , two of are prime.
Theorem (GPY, 2005). Assume EH() for some , and let be admissible, where . Then, for infinitely many , two of are prime.
With Zhang’s work, this shows that improvements in lead to improvements in ; roughly speaking . In fact it is clear that is bounded above by the diameter of any admissible -tuple. Consequently, a lot of effort is being devoted towards the task of finding narrow admissible tuples.
We conclude with a sketch of the proof of the above Theorem of GPY. Define
where is the indicator function of being almost prime. The idea is that if then, reasonably often, two of would be prime. It remains to estimate the numerator and denominator of . To this end, we introduce Selberg’s functions which, with judicious weights, approximate our previous . Fix with , and put , where
where the are real weights with . Note that majorises the indicator function of the primes, for if is prime and then .
First consider the denominator of . Our weights can be chosen so that, as well as the two notions of being roughly compatible, we can show that
The derivation of (2) proceeds as follows. Note that
where denotes the least common multiple. If then the inner sum is roughly , so
The right hand sum is a quadratic form in , which can be minimised using general theory of the Selberg sieve, leading to the inequality (2).
The numerator of is amenable to a similar treatment. The attendant sum
can be controlled providing that (1) holds with in place of . A painful calculation gives
and this exceeds (for large ) if .