Exponential sums over integers without large prime divisors

Abstract

We obtain a new bound on exponential sums over integers without large prime divisors, improving that of Fouvry and Tenenbaum (1991). For a fixed integer $\hat{I}½\ne 0$, we also obtain new bounds on exponential sums with $\hat{I}½$-th powers of such integers. The improvement is based on exploiting more precisely the factorisation of integers without large prime divisors, along with existing Type~I and Type~II bounds. For $\hat{I}½=1$ we use the classical bounds of Vinogradov (1937), while for $\hat{I}½\ne 1$ we use bounds of Vaughan (1975) as well as of Fouvry, Kowalski and Michel (2014).

New version: It has a new co-author, uses a different approach and gives a substantial improvement of results from the previous version