A Ces\`aro average for an additive problem with an arbitrary number of prime powers and squares

TL;DR

This paper advances the understanding of representing integers as sums of prime powers and squares by extending previous weighted average results with Cesàro weights, achieving optimal outcomes with the current methods.

Contribution

It generalizes and improves existing results on Cesàro averages for sums involving prime powers and squares, covering all previously studied cases and optimizing the technique.

Findings

Unified framework for weighted averages of representations

Improved bounds for sums involving prime powers and squares

Achieved optimal results within the chosen methodological approach

Abstract

In this paper we extend and improve all the previous results known in literature about weighted average, with Ces\`aro weight, of representations of an integer as sum of a positive arbitrary number of prime powers and a non-negative arbitrary number of squares. Our result includes all cases dealt with so far and allows us to obtain the best possible outcome using the chosen technique.