A Cesàro Average of Goldbach numbers

Abstract

Let $\Lambda$ be the von Mangoldt function and $(r_G(n)={\sum }_{m_1+m_2=n}\Lambda (m_1)\Lambda (m_2))$ be the counting function for the Goldbach numbers. Let $N\ge 2$ be an integer. We prove that $$\begin{array}{rl}& \sum _{n\le N}{r}_G(n)\frac{(1-n/N{)}^k}{\Gamma (k+1)}=\frac{N^2}{\Gamma (k+3)}-2\sum _{\rho }\frac{\Gamma (\rho )}{\Gamma (\rho +k+2)}{N}^{\rho +1}\\ & +\sum _{{\rho }_1}\sum _{{\rho }_2}\frac{\Gamma ({\rho }_1)\Gamma ({\rho }_2)}{\Gamma ({\rho }_1+{\rho }_2+k+1)}{N}^{{\rho }_1+{\rho }_2}+{\mathcal{O}}_k(N^{1/2}),\end{array}$$ for $k>1$, where $\rho$, with or without subscripts, runs over the non-trivial zeros of the Riemann zeta-function $\zeta (s)$.

arXiv commentsubmitted

References in corpus (2)

• The number of Goldbach representations of an integer
• Sums of many primes

Cited by in corpus (5)

• Sum of one prime and two squares of primes in short intervals
• A Cesàro Average of Hardy-Littlewood numbers
• On an Average Goldbach Representation Formula of Fujii
• A Cesàro average for an additive problem with an arbitrary number of prime powers and squares
• The average number of Goldbach representations over multiples of $q$