Explicit formula for the average of goldbach numbers

Let $Lambdaleft(n ight)$ be the Von Mangoldt function, let [r_{G}left(n ight):=\underset{{scriptstyle m_{1}+m_{2}=n}}{sum_{m_{1},m_{2}leq n}}Lambdaleft(m_{1} ight)Lambdaleft(m_{2} ight),] be the counting function of the Goldbach numbers and the counting function of the prime tuples, respectively. Let $N>2$ be an integer. We will find the explicit formulae for the average of $r_{G}left(n ight)$ in terms of elementary functions, the incomplete Beta function $B_{z}left(a,b ight)$,series over $ ho$ that, with or without subscript, runs over the non-trivial zeros of the Riemann Zeta function and the Dilogarithm function. We will also prove the explicit formulae in an asymptotic form and a truncated formula for the average of $r_{G}left(n ight)$. Some observation about these formulae and the average with Cesàro weight [ rac{1}{Gammaleft(k+1 ight)}sum_{nleq N}r_{G}left(n ight)left(N-n ight)^{k},,k>0] and [r_{PT}left(N,h ight):=sum_{n=0}^{N}Lambdaleft(n ight)Lambdaleft(n+h ight),,hinmathbb{N}] are included.