A Programme for the Twin Prime Problem via Quadratic Bilinear Dispersion

AbstractWe formulate a precise analytic reduction of the twin prime problem to a single,localised dispersion inequality (Theorem X ) for the two-point correlation of the Liouvillefunction. The route combines smooth weights with a roughness filter, a two-dimensionalfundamental lemma of the sieve to count candidate pairs, and a proof of strong decayfor twisted short sums via contour shifting and a classical zero-free region. AssumingTheorem X, we obtain a lower bound ≫ x/( log x)2 for the number of twin prime pairsup to x, and hence infinitely many twin primes. Theorem X is the deliberately isolatedtarget; we document two independent strategies towards it: an additive-energy/incidenceroute and a decoupling-transfer route.