On the Cesà ro average of the numbers that can be written as sum of a prime and two squares of primes

Let Λ(n) be the Von Mangoldt function and rSP(n)=∑m1+m22+m32=nΛ(m1)Λ(m2)Λ(m3) be the counting function for the numbers that can be written as sum of a prime and two squares. Let N be a sufficiently large integer. We prove that∑n≤NrSP(n)(N−n)kΓ(k+1)=Nk+2π4Γ(k+3)+E(N,k) for k>3/2, where E(N,k) consists of lower order terms that are given in terms of k and sum over the non-trivial zeros of the Riemann zeta function.