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arXiv · 2404.10278

Exponential sums over integers without large prime divisors

Sary Drappeau·Igor E. Shparlinski
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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 $ν\ne 0$, we also obtain new bounds on exponential sums with $ν$-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 $ν=1$ we use the classical bounds of Vinogradov (1937), while for $ν\neq 1$ we use bounds of Vaughan (1975) as well as of Fouvry, Kowalski and Michel (2014).

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Sary Drappeau, Igor E. Shparlinski. 2025-05-05. Exponential sums over integers without large prime divisors. https://arxiv.org/abs/2404.10278

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