A Cesàro Average of Hardy-Littlewood numbers

Abstract

Let $Λ$ be the von Mangoldt function and $r_{\textit{HL}}(n) = \sum_{m_1 + m_2^2 = n} Λ(m_1),$ be the counting function for the Hardy-Littlewood numbers. Let $N$ be a sufficiently large integer. We prove that $$\begin{align}\sum_{n \le N} r_{\textit{HL}}(n) \frac{(1 - n/N)^k}{Γ(k + 1)} &= \frac{π^{1 / 2}}2 \frac{N^{3 / 2}}{Γ(k + 5 / 2)} - \frac 12 \frac{N}{Γ(k + 2)} - \frac{π^{1 / 2}}2 \sum_ρ \frac{Γ(ρ)}{Γ(k + 3 / 2 + ρ)} N^{1 / 2 + ρ}\\ &+ 1/2 \sum_ρ \frac{Γ(ρ)}{Γ(k + 1 + ρ)} N^ρ + \frac{N^{3 / 4 - k / 2}}{π^{k + 1}} \sum_{\ell \ge 1} \frac{J_{k + 3 / 2} (2 π\ell N^{1 / 2})}{\ell^{k + 3 / 2}}\\ &- \frac{N^{1 / 4 - k / 2}}{π^k} \sum_ρ Γ(ρ) \frac{N^{ρ/ 2}}{π^ρ} \sum_{\ell \ge 1} \frac{J_{k + 1 / 2 + ρ} (2 π\ell N^{1 / 2})} {\ell^{k + 1 / 2 + ρ}} + \mathcal{O}_k(1).\end{align}$$ 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 $u$.