Weyl sums, mean value estimates, and Waring's problem with friable numbers

TL;DR

This paper investigates Weyl sums over friable integers, providing asymptotic formulas, bounds, and mean value estimates, and applies these results to Waring's problem with friable numbers, extending classical results to this special subset.

Contribution

It introduces new estimates and formulas for Weyl sums over friable integers and applies them to Waring's problem, extending classical analytic number theory techniques to friable numbers.

Findings

Asymptotic formula for Weyl sums in major arcs

Nontrivial bounds for Weyl sums in minor arcs

Mean value estimates for friable Weyl sums

Abstract

In this paper we study Weyl sums over friable integers (more precisely $y$-friable integers up to $x$ when $y = (\log x)^C$ for a large constant $C$). In particular, we obtain an asymptotic formula for such Weyl sums in major arcs, nontrivial upper bounds for them in minor arcs, and moreover a mean value estimate for friable Weyl sums with exponent essentially the same as in the classical case. As an application, we study Waring's problem with friable numbers, with the number of summands essentially the same as in the classical case.