Abstract
Let Λ(n) be the von Mangoldt function, and let rG(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 S(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 S(z) and the Cesaro average of rG(n).
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