On Waring's problem in sums of three cubes

TL;DR

This paper studies the number of ways large integers can be expressed as sums of three positive cubes, focusing on asymptotic formulas and bounds for such representations.

Contribution

It introduces new asymptotic formulas and lower bounds for representations of integers as sums of three cubes, extending understanding of Waring's problem.

Findings

Derived asymptotic formulas for representations as sums of three cubes.

Established lower bounds for the count of such representations.

Analyzed the multiplicity aspects in representations.

Abstract

We investigate the asymptotic formula for the number of representations of a large positive integer as a sum of $k$-th powers of integers represented as the sums of three positive cubes, counted with multiplicities. We also obtain a lower bound for the number of representations when the sums of three cubes are counted without multiplicities.