A Cesàro average for an additive problem with prime powers

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. Let $1\le {\ell }_1\le {\ell }_2$ be two integers, $\Lambda$ be the von Mangoldt function and % \(r_{\ell_1,\ell_2}(n) = \sum_{m_1^{\ell_1} + m_2^{\ell_2}= n} Λ(m_1) Λ(m_2) \) % be the weighted counting function for the number of representation of an integer as a sum of two prime powers. Let $N\ge 2$ be an integer. We prove that the Cesàro 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.

Accepted (Mar. 2018) for publication in the Proceedings of the conference "Number Theory Week", Poznan, September 4-8, 2017. One reference updated. arXiv admin note: substantial text overlap with arXiv:1206.0251