A Ces\`aro average for an additive problem with prime powers

TL;DR

This paper extends results on weighted averages for representations of integers as sums of two prime powers, providing explicit formulas involving the zeros of the Riemann zeta-function.

Contribution

It introduces an improved Cesàro average analysis for the number of representations of integers as sums of two prime powers, explicitly involving zeta zeros.

Findings

Cesàro average of weighted representations expressed via zeta zeros

Explicit development for sums of two prime powers

Enhanced understanding of prime power additive problems

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 % \(r_{\ell_1,\ell_2}(n) = \sum_{m_1^{\ell_1} + m_2^{\ell_2}= n} \Lambda(m_1) \Lambda(m_2) \) % 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.