Weyl sums, mean value estimates, and Waring's problem with friable numbers
Volume 176 / 2016
Acta Arithmetica 176 (2016), 249-299
MSC: Primary 11L07; Secondary 11P05, 11N25.
DOI: 10.4064/aa8448-7-2016
Published online: 17 October 2016
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, non-trivial 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.
