Resumen de Generalising the Hardy�Littlewood method for primes

Benjamin Green

• The Hardy�Littlewood method is a well-known technique in analytic number theory.

  Among its spectacular applications are Vinogradov�s 1937 result that every sufficiently large odd number is a sum of three primes, and a related result of Chowla and Van der Corput giving an asymptotic for the number of 3-term progressions of primes, all less than N. This article surveys recent developments of the author and T. Tao, in which the Hardy�Littlewood method has been generalised to obtain, for example, an asymptotic for the number of 4-term arithmetic progressions of primes less than N.