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--- abstract: | Let $\Lambda\left(n\right)$ be the von Mangoldt function and $r_{G}\left(n\right):=\sum_{m_{1}+m_{2}=n}\Lambda\left(m_{1}\right)\Lambda\left(m_{2}\right)$ be the counting function for the numbers that can be written as sum of two primes (that we will call “Goldbach numbers”, for brevity) and let $\widetilde{S}\left(z\right):=\sum_{n\geq1}\Lambda\left(n\right)e^{-nz}$, with $z\in\mathbb{C}$, $\mathrm{Re}\left(z\right)>0$. In this paper we will prove the identity $$\begin{aligned} \widetilde{S}\left(z\right)= & 2e^{-2z}+\frac{e^{-2z}}{z}-\sum_{\rho}z^{-\rho}\Gamma\left(\rho\right)\\ + & \sum_{\rho}\left(z^{-\rho}\gamma\left(\rho,2z\right)-\frac{2^{\rho}e^{-z}}{\rho}\right)-\frac{\zeta^{\prime}}{\zeta}\left(0\right)e^{-2z}\\ + & \frac{e^{-z}}{2}\left(-i\pi\textrm{sgn}\left(\textrm{Im}\left(-z\right)\right)+\textrm{Ei}\left(-z\right)\right)\\ + & \frac{e^{z}}{2}\left(-i\pi\textrm{sgn}\left(\textrm{Im}\left(-z\right)\right)-\log\left(3\right)e^{-3z}+\textrm{Ei}\left(-3z\right)\right)\\ + & i\pi\textrm{sgn}\left(\textrm{Im}\left(-z\right)\right)+\log\left(2\right)e^{-2z}-\textrm{Ei}\left(-2z\right)\end{aligned}$$ where $\gamma\left(\rho,2z\right)$ is the lower incomplete Gamma function, $\rho=\beta+i\gamma$ runs over the non-trivial zeros of the Riemann Zeta function and $$\mathrm{Ei}\left(z\right)=-\int_{-z}^{\infty}\frac{e^{-t}}{t}dt,\ \left|\arg\left(z\right)\right|<\pi$$ is the Exponential integral function. In addition we will prove that $$\begin{aligned} \sum_{n\leq N}r_{G}\left(n\right)\left(N-n\right)= & \frac{N^{3}}{6}-2\sum_{\rho}\frac{\left(N-2\right)^{\rho+2}}{\rho\left(\rho+1\right)\left(\rho+2\right)}\\ + & \sum_{\rho_{1}}\sum_{\rho_{2}}\frac{\Gamma\left(\rho_{1}\right)\Gamma\left(\rho_{2}\right)}{\Gamma\left(\rho_{1}+\rho_{2}+2\right)}N^{\rho_{1}+\rho_{2}+1}-2\sum_{\rho_{1}}\sum_{\rho_{2}}\frac{N^{\rho_{1}+\rho_{2}+1}}{\rho_{1}\rho_{2}}B_{2/N}\left(\rho_{1}+1,\rho_{2}+1\right)\\ + & F\left(N\right)\end{aligned}$$ where $N>4$ is a natural number, $B_{x}\left(a,b\right)$ is the incomplete Beta function and $F\left(N\right)$ is a sum of (explicitly calculate) elementary functions, dilogarithms and sums over non-trivial zeros of the Riemann Zeta function involving the incomplete Beta function. Furthemore we will prove that $$F\left(N\right)=O\left(N^{2}\right)$$ as $N\rightarrow\infty.$ author: - Marco Cantarini title: Some identities involving the Cesàro average of Goldbach numbers --- 2010 Mathematics Subject Classification:[ Primary 11P32; Secondary 44A10]{} Introduction ============ In this paper we prove some identities inherent the study to Cesàro average of Goldbach numbers. The main result in this direction is the paper of Languasco e Zaccagnini [@Langzac]. The authors proved the formula $$\begin{aligned} \sum_{n\leq N}r_{G}\left(n\right)\frac{\left(N-n\right)^{k}}{\Gamma\left(k+1\right)}= & \frac{N^{2+k}}{\Gamma\left(k+3\right)}-2\sum_{\rho}\frac{\Gamma\left(\rho\right)}{\Gamma\left(\rho+k+2\right)}N^{\rho+k+1}\nonumber \\ + & \sum_{\rho_{1}}\sum_{\rho_{2}}\frac{\Gamma\left(\rho_{1}\right)\Gamma\left(\rho_{2}\right)}{\Gamma\left(\rho_{1}+\rho_{2}+k+1\right)}N^{\rho_{1}+\rho_{2}+k}+O_{k}\left(N^{k+1}\right)\label{langzac formula}\end{aligned}$$ for $k>1$, where $N$ is a large natural number and $\rho$, with or without subscripts, runs over the non-trivial zeros of the Riemann Zeta function $\zeta\left(s\right)$. The parameter $k$ plays a central role in the study of these problems and we would like to keep it as small as possible since, for $k=0$, the Cesàro weight disappears. The method of Languasco and Zaccagnini allows to study many types of additive problems; see [@Canta1], [@Canta2] and [@LangZac2]. The crucial points of their technique are the relation $$\sum_{n\leq N}r_{G}\left(n\right)\frac{\left(N-n\right)^{k}}{\Gamma\left(k+1\right)}=\frac{1}{2\pi i}\int_{\left(1/N\right)}e^{Nz}z^{-k-1}\widetilde{S}\left(z\right)^{2}dz\label{identit=0000E0 fondamentale}$$ which holds for $k>0,$ where $$\widetilde{S}\left(z\right):=\sum_{n\geq1}\Lambda\left(n\right)e^{-nz},\,z=a+iy,\,\mathrm{with}\,a>0,\,y\in\mathbb{R}.$$ is the power series of the Von Mangoldt function and $\int_{\left(1/N\right)}$ means $\int_{1/N-i\infty}^{1/N+i\infty}$, and the estimation $$\widetilde{S}\left(z\right)=\frac{1}{z}-\sum_{\rho}z^{-\rho}\Gamma\left(\rho\right)+E\left(a,y\right)\label{eq:asint s tilda zac}$$ where $$E\left(a,y\right)\ll\begin{cases} 1+\left|z\right|^{1/2}, & \left|y\right|\leq a\\ 1+\left|z\right|^{1/2}\left(1+\log^{2}\left(\left|y\right|/a\right)\right), & \left|y\right|>a. \end{cases}\label{errore s tilda}$$ The proof of is present in [@Langzac]. We recall that the study of $\widetilde{S}\left(z\right)$ is classical, see for example [@HardLit]. One of the Recently Goldston and Yang [@GolYan] proved, assuming RH, that $$\sum_{n\leq N}r_{G}\left(n\right)\left(1-\frac{n}{N}\right)=\frac{N^{2}}{6}-2\sum_{\rho}\frac{\Gamma\left(\rho\right)}{\Gamma\left(\rho+3\right)}N^{\rho+1}+O\left(N\right)\label{eq:GoldeYan}$$ which correspond to what Languasco and Zaccagnini’s formula implies if one take $k=1$. Languasco and Zaccagnini [@LangZacNew] also proved (assuming RH) that $$\begin{aligned} \sum_{n=N-H}^{N+H}r_{G}\left(n\right)\left(1-\frac{\left|n-N\right|}{H}\right)= & HN-\frac{2}{H}\sum_{\rho}\frac{\left(N+H\right)^{\rho+2}-2N^{\rho+2}+\left(N-H\right)^{\rho+2}}{\rho\left(\rho+1\right)\left(\rho+2\right)}\\ + & O\left(N\log^{2}\left(\frac{2N}{H}\right)+H\log^{2}\left(N\right)\log\left(2H\right)\right)\end{aligned}$$ where $N\geq2,\,1\leq H\leq N$. $k.$ In a very recent paper Brüdern, Kaczorowski and Perelli [@Perelli] were able to find an explicit formula which holds for all $k>0.$ Similar averages of arithmetical functions are common in literature, see, e.g., Chandrasekharan - Narasimhan [@Chana] and Berndt [@Ber] who built on earlier classical work. In In this paper we will prove an explicit formula for $\widetilde{S}\left(z\right)$ which will allow us to find a smaller error respect to . Kunik and Lucht [@Kunik] wrote an explicit formula for