Some Identities Involving the Cesàro Average of the Goldbach Numbers

Resumo

Let Λ(n) be the von Mangoldt function, and let r_G(n):= ∑_m1+m2=n Λ (m₁)Λ(m₂) be the weighted sum for the number of Goldbach representations which also includes powers of primes. Let S̃(z): = ∑_n≥1 Λ (n)e-nz, where Λ (n) is the Von Mangoldt function, with z ∈ ℂ, Re (z) > 0. In this paper, we prove an explicit formula for S̃(z) and the Cesàro average of r_G(n).