Cesàro average in short intervals for Goldbach numbers

Abstract:

Let $\Lambda$ be the von Mangoldt function and \begin{equation*} R(n) = \sum _{h+k=n} \Lambda (h)\Lambda (k). \end{equation*} Let further $N,H$ be two integers, $N\ge 2$, $1\le H \le N$, and assume that the Riemann Hypothesis holds. Then \begin{align*} \sum _{n=N-H}^{N+H} R(n) \Bigl (1- \frac {\vert n- N \vert }{H}\Bigr ) &= HN -\frac {2}{H} \sum _{\rho } \frac {(N+H)^{\rho +2} - 2 N^{\rho +2} +(N-H)^{\rho +2} }{\rho (\rho + 1)(\rho + 2)} \\& + \mathcal {O}\Bigl ({N \Bigl (\log \frac {2N}{H}\Bigr )^2 + H (\log N)^2 \log (2H) }\ , \end{align*} where $\rho =1/2+i\gamma$ runs over the non-trivial zeros of the Riemann zeta function $\zeta (s)$.