In this paper we extend and improve our results on weighted averages for the number of representations of an integer as a sum of two powers of primes. Let $1\le \ell_1 \le \ell_2$ be two integers, $\Lambda$ be the von Mangoldt function and % \(r_{ \ell_1,\ell_2}(n) = \sum_{ m_1^{ \ell_1} + m_2^{ \ell_2}= n} \Lambda(m_1) \Lambda(m_2) \) % be the weighted counting function for the number of representation of an integer as a sum of two prime powers. Let $N \geq 2$ be an integer. We prove that the Ces\`aro average of weight $k > 1$ of $r_{ \ell_1,\ell_2}$ over the interval $[1, N]$ has a development as a sum of terms depending explicitly on the zeros of the Riemann zeta-function.
Categories: math.NT