A Ces\`aro Average of Hardy-Littlewood numbers

Let $\mathrm{\Lambda }$ be the von Mangoldt function and $r_{$\mathit{\text{HL}}$}(n) = \sum_{m_1 + m_2^2 = n} \Lambda(m_1),$ 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_{$\mathit{\text{HL}}$}(n) \frac{(1 - n/N)^k}{\Gamma(k + 1)} &= \frac{\pi^{1 / 2}}2 \frac{N^{3 / 2}}{\Gamma(k + 5 / 2)} - \frac 12 \frac{N}{\Gamma(k + 2)} - \frac{\pi^{1 / 2}}2 \sum_{\rho} \frac{\Gamma(\rho)}{\Gamma(k + 3 / 2 + \rho)} N^{1 / 2 + \rho}\ &+ 1/2 \sum_{\rho} \frac{\Gamma(\rho)}{\Gamma(k + 1 + \rho)} N^{\rho} + \frac{N^{3 / 4

• k / 2}}{\pi^{k + 1}} \sum_{\ell \ge 1} \frac{J_{k + 3 / 2} (2 \pi \ell N^{1 / 2})}{\ell^{k + 3 / 2}}\ &- \frac{N^{1 / 4 - k / 2}}{\pi^k} \sum_{\rho} \Gamma(\rho) \frac{N^{\rho / 2}}{\pi^\rho} \sum_{\ell \ge 1} \frac{J_{k + 1 / 2

• \rho} (2 \pi \ell N^{1 / 2})} {\ell^{k + 1 / 2 + \rho}} + $\mathcal{O}$_k(1).\end{align}$$for$k > 1$,where$\rho$ runs over the non-trivial zeros of the Riemann zeta-function $\zeta (s)$ and $J_{\nu }(u)$ denotes the Bessel function of complex order $\nu$ and real argument $u$.