On a variance associated with the distribution of $r_3(n)$ in arithmetic progressions

TL;DR

This paper investigates the variance in the distribution of the function r_3(n), which counts representations of n as a sum of three positive cubes, within arithmetic progressions, using Hardy-Littlewood and Farey sequence techniques.

Contribution

It combines the study of variance in arithmetic progressions with the analysis of r_3(n), providing new insights into their joint behavior and distribution.

Findings

Derived new asymptotic formulas for the variance of r_3(n) in progressions.

Applied Hardy-Littlewood method and Farey sequences to analyze distribution patterns.

Extended previous results on r_3(n) to the context of arithmetic progressions.

Abstract

There are two questions in analytic number theory which have attracted much attention over the years. The first one is about the asymptotic formula for the variance associated with the distribution of a real sequence in arithmetic progressions, which origins in the work of Barban. The second one is about the function $r_3(n)$, the number of ordered representation of $n$ in the sum of three positive cubes, and several results are recently proved by Robert C. Vaughan. The paper is concerned with the conjunction of these questions, with the application of the Hardy-Littlewood Method and the Farey sequence.