Additive problems with almost prime squares

Contact: Pieter Moree

The prototype equation of this talk is n = p + x^2 + y^2, considered first by Hardy and Littlewood and later by Hooley and Linnik. It is shown that this equation and variations of it have the expected number of representations with x and y being almost primes (i.e. having a bounded number of prime factors).The proof involves a mix of algebraic number theory, algebraic geometry to estimate multiple exponential sums, automorphic forms and analytic number theory. The method can also handle the case when x and y are smooth numbers. This is joint work with L. Grimmelt, J. Li and S. Myerson.