The asymptotic formula in Waring's problem: higher order expansions

TL;DR

This paper derives a detailed multi-term asymptotic expansion for counting the representations of large numbers as sums of k-th powers, advancing understanding in Waring's problem for large s and k.

Contribution

It provides the first explicit higher-order asymptotic formulas for Waring's problem when s is sufficiently large relative to k.

Findings

Explicit multi-term asymptotic expansion derived

Improves precision in counting representations for large numbers

Advances theoretical understanding of Waring's problem

Abstract

When k > 1 and s is sufficiently large in terms of k, we derive an explicit multi-term asymptotic expansion for the number of representations of a large natural number as the sum of s positive integral k-th powers.