On Lagrange's theorem with prime variables

It is conjectured that Lagrange's theorem on four squares is true for prime variables, that is, every large integer n with n equivalent to 4 (mod 24) is the sum of four squares of primes. In this paper, the size for the exceptional set in the above conjecture is reduced to O(N2/5+epsilon). The new ingredients include an iterative method to treat the enlarged major arcs, and a hybrid estimate for Dirichlet polynomials.