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 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 rG(n).
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Acknowledgments
I thank my mentor Alessandro Zaccagnini, as well as Jacopo “Jack” D’Aurizio and Matthias Kunik, for several conversations on this topic. I would like to thank the referee for his remarks about this work.
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Cantarini, M. Some Identities Involving the Cesàro Average of the Goldbach Numbers. Math Notes 106, 688–702 (2019). https://doi.org/10.1134/S0001434619110038
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DOI: https://doi.org/10.1134/S0001434619110038