A Cesàro average for an additive problem with prime powers
Volume 118 / 2019
Abstract
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 (the paper of the authors in Forum Math. 27 (2015), see also the paper of A.L., Riv. Mat. Univ. di Parma 7 (2016), Theorem 2.2). 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 Cesàro average of weight $k \gt 1$ 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.
