Asymptotics for some polynomial patterns in the primes

We prove asymptotic formulae for sums of the form where K is a convex body, each F_i is either the von Mangoldt function or the representation function of a quadratic form, and = (₁, ...,_t) is a system of linear forms of finite complexity. When all the functions F_i are equal to the von Mangoldt function, we recover a result of Green and Tao, while when they are all representation functions of quadratic forms, we recover a result of Matthiesen. Our formulae imply asymptotics for some polynomial patterns in the primes. For instance, they describe the asymptotic behaviour of the number of k-term arithmetic progressions of primes whose common difference is a sum of two squares.The paper combines ingredients from the work of Green and Tao on linear equations in primes and that of Matthiesen on linear correlations amongst integers represented by a quadratic form. To make the von Mangoldt function compatible with the representation function of a quadratic form, we provide a new pseudorandom majorant for both - an average of the known majorants for each of the functions - and prove that it has the required pseudorandomness properties.