Bombieri–Friedlander–Iwaniec dispersion-method proof
The argument
Bombieri, Friedlander and Iwaniec apply Linnik's dispersion method to the error terms of primes in progressions weighted by a well-factorable sequence, and because such a sequence factors at any chosen scale the resulting bilinear forms can be arranged so that their off-diagonal terms are sums of Kloosterman sums; given the Deshouillers–Iwaniec bounds for sums of Kloosterman sums, those terms are small for moduli up to x^{4/7-ε}, which yields the theorem.
Does it work?
Not yet evaluated.
Premises
- UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →The Deshouillers–Iwaniec bounds for sums of Kloosterman sums hold.
Where it is made
Passages where sources make, criticize or mention this argument. They record where it appears, not how strong it is, and none has precedence over another.
No source is recorded as making it.
Earlier ratings
- Sep 16, 2026
The inference goes through: the dispersion method reduces the weighted error terms to bilinear forms whose off-diagonal contribution is a sum of Kloosterman sums, and the factorability of the weights lets the scales be chosen so that the Kloosterman bounds win up to exponent 4/7. The argument lives on the Deshouillers–Iwaniec bounds for sums of Kloosterman sums, a standard refereed result in continuous use since 1982, so nothing in its premises is in doubt. The proof was refereed in Acta Mathematica in 1986 and has been independently expounded and reused since, most recently as the starting point of Maynard's and Lichtman's extensions.