A Cesàro average for Hardy-Littlewood numbers

Let Λ be the von Mangoldt function and rHL(n)=∑m1+m22=nΛ(m1) be the counting function for the Hardy-Littlewood numbers. Let N be a sufficiently large integer. We prove that ∑n≤NrHL(n)(1n/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+N3/4k/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>1, where ρ runs over the non-trivial zeros of the Riemann zeta-function ζ(s) and Jν(u) denotes the Bessel function of complex order ν and real argument u. © 2012 Elsevier Ltd.