On the Sums of three or more primes in arithmetic progressions

As one application of circle method,considering the equation p1+p2 +…+pk=N with pj≡gj(modh),j =1,2,…,k,∑1≤j≤kgj≡N(modh),k≥3,a representable asymptotic formula under large modulo was given.That is,suppose k≥3,Θ=sup{β:L(β+iγ,χ)=0},e0,1≤h≤Nδ,0δ1,then ∑ p1+p2+…+pk=N pj≤N,pj≡gj(modh),1≤j≤k (logp1)(logp2)…(logpk)=1(k-1)!Nk-1G(k,N)+O(Nk-2++e)+O(Nηk+e)+O(Nk-2+λ+e),where η3=5/9,η4=5/14 and ηk=0(k≥5),λ=,if L function exists exceptional zero ,0,otherwise,G(k,N)=hφ(h)k∏ph,pN1+(-1)k+1(p-1)k∏ph,p|N1+(-1)k(p-1)k-1.