A Cesàro average of Hardy–Littlewood numbers

Let Λ be the von Mangoldt function and r H L ( n ) = ∑ m 1 + m 2 2 = n Λ ( m 1 ) be the counting function for the Hardy–Littlewood numbers. Let N be a sufficiently large integer. We prove that ∑ n ≤ N r H L ( n ) ( 1 − n / N ) k Γ ( k + 1 ) = π 1 / 2 2 N 3 / 2 Γ ( k + 5 / 2 ) − 1 2 N Γ ( k + 2 ) − π 1 / 2 2 ∑ ρ Γ ( ρ ) Γ ( k + 3 / 2 + ρ ) N 1 / 2 + ρ + 1 2 ∑ ρ Γ ( ρ ) Γ ( k + 1 + ρ ) N ρ + N 3 / 4 − k / 2 π k + 1 ∑ ℓ ≥ 1 J k + 3 / 2 ( 2 π ℓ N 1 / 2 ) ℓ k + 3 / 2 − N 1 / 4 − k / 2 π k ∑ ρ Γ ( ρ ) N ρ / 2 π ρ ∑ ℓ ≥ 1 J k + 1 / 2 + ρ ( 2 π ℓ N 1 / 2 ) ℓ k + 1 / 2 + ρ + O k ( 1 ) , for k > 1 , where ρ runs over the non-trivial zeros of the Riemann zeta-function ζ ( s ) and J ν ( u ) denotes the Bessel function of complex order ν and real argument u .