Sums of three or more primes in arithmetical progressions

As one application of circle method, the object of this paper is to consider the equation (p_1+)p_2+…+p_k=N with p_j≡g_j(mod()h), j=1,2,…,k, ∑_(1≤j≤k)g_j≡N(mod()h), k≥3, and to give a representable asymptotic formula by means of the method of FRIEDLANDER and GOLDSTON. That is, suppose k≥3, Θ=sup{β:L(β+iγ)=0}, e0, h is a given positive integer, then ∑_(p_1+p_2+…+p_k=N,p_j≤N,p_j≡g_j(mod h),1≤j≤k)(ln()p_1)(ln()p_2)·…·(ln()p_k)= ((k-1)!)~(-1)N~(k-1)G(k,N)+O(N~(k-2+Θ+e)+N~(η_k+e)), where G(k,N) is the singular series, η_3=9/5, η_4=13/5, η_k=0 (k≥5).