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A Cesàro average for an additive problem with an arbitrary number of prime powers and squares

Marco Cantarini, Alessandro Gambini, Alessandro Zaccagnini

Abstract

In this paper we extend and improve all the previous results known in literature about weighted average, with Cesàro 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.

A Cesàro average for an additive problem with an arbitrary number of prime powers and squares

Abstract

In this paper we extend and improve all the previous results known in literature about weighted average, with Cesàro 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.

Paper Structure

This paper contains 10 sections, 4 theorems, 107 equations.

Table of Contents

  1. introduction
  2. preliminary definitions and main theorem
  3. Settings
  4. Lemmas
  5. proof of the main theorem
  6. Error term
  7. Evaluation of the main term
  8. Evaluation of $I_{1}$
  9. Evaluation of $I_{2}$
  10. Evaluation of $I_{3}$

Key Result

Theorem 1

Let $d,h\in\mathbb{N},\,d>0,$ let $N$ be a sufficiently large integer. Let $\mathfrak{D}:=\left\{ 1,\dots,d\right\}$ and, for every $\mathfrak{J}\subseteq\mathfrak{D}$ (or with the notation $I\subseteq\mathfrak{D})$ let $\tau\left(\mathbf{r},\mathfrak{J}\right):=\sum_{j\in\mathfrak{J}}\frac{1}{r_{j}

Theorems & Definitions (6)

  • Theorem 1
  • Lemma 2
  • Lemma 3
  • proof
  • Lemma 4
  • proof