Аннотация:
Let
$\Lambda(n)$
be the von Mangoldt function, and let
$r_{G}(n):=\sum_{m_{1}+m_{2}=n}\Lambda(m_{1})\Lambda(m_{2})$
be the
weighted sum for the number of Goldbach representations
which also includes powers of primes.
Let
$\widetilde{S}(z):=\sum_{n\geq1}\Lambda(n)e^{-nz}$,
where
$\Lambda(n)$
is the
Von Mangoldt function, with
$z\in\mathbb{C}, \mathrm{Re}(z)>0$.
In this paper,
we prove an explicit formula for
$\widetilde{S}(z)$
and the Cesàro
average of
$r_{G}(n)$.