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

TL;DR

This paper extends previous work to analyze the asymptotic behavior of the average number of representations of integers as sums of two prime powers over multiples of a fixed integer, providing new insights into additive number theory.

Contribution

It introduces a generalized asymptotic formula for counting representations of integers as sums of two prime powers over multiples of a fixed integer, expanding prior results.

Findings

Derived asymptotic formulas for the average number of representations

Extended previous results to prime powers with k ≥ 2

Provided new bounds and estimates for the counting functions

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 $k\ge 2$.