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The average number of representations of an integer as a sum of two prime powers over multiples of a fixed integer

Alessandra Migliaccio, Alessandro Zaccagnini

Abstract

We extend a result by Ikeda and Suriajaya (2025) to find the asymptotic behaviour of the average number of representations of an integer $n$, over multiples of a fixed $q\ge 2$, as a sum of two prime $k$-th powers, for $k\ge 2$.

The average number of representations of an integer as a sum of two prime powers over multiples of a fixed integer

Abstract

We extend a result by Ikeda and Suriajaya (2025) to find the asymptotic behaviour of the average number of representations of an integer , over multiples of a fixed , as a sum of two prime -th powers, for .
Paper Structure (9 sections, 6 theorems, 82 equations)

This paper contains 9 sections, 6 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Lemmas and sketch of the proof
  3. Proof of Theorem \ref{['th2']}
  4. Evaluation of I3
  5. Evaluation of I1
  6. Evaluation of I2
  7. Evaluation of I1NH
  8. Evaluation of I2NH
  9. Evaluation of Sigma(q)

Key Result

Theorem 1.1

Assume that GRH holds for Dirichlet L-functions $L(s,\chi)$ associated with characters $\chi\bmod q$. For $2\le q\le N$, we have

Theorems & Definitions (20)

  • Theorem 1.1: Theorem 1.3 of ike25
  • Definition 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Example 1.1
  • Remark 1.6
  • Remark 1.7
  • Lemma 2.1
  • ...and 10 more