A Cesàro average of Hardy–Littlewood numbers

Let ΛΛ be the von Mangoldt function and rHL(n)=∑m1+m22=nΛ(m1) be the counting function for the Hardy–Littlewood numbers. Let NN be a sufficiently large integer. We prove that ∑n≤NrHL(n)(1−n/N)kΓ(k+1)=π1/22N3/2Γ(k+5/2)−12NΓ(k+2)−π1/22∑ρΓ(ρ)Γ(k+3/2+ρ)N1/2+ρ+12∑ρΓ(ρ)Γ(k+1+ρ)Nρ+N3/4−k/2πk+1∑ℓ≥1Jk+3/2(2πℓN1/2)ℓk+3/2−N1/4−k/2πk∑ρΓ(ρ)Nρ/2πρ∑ℓ≥1Jk+1/2+ρ(2πℓN1/2)ℓk+1/2+ρ+Ok(1), for k>1k>1, where ρρ runs over the non-trivial zeros of the Riemann zeta-function ζ(s)ζ(s) and Jν(u)Jν(u) denotes the Bessel function of complex order νν and real argument uu.