Local Fourier uniformity of higher divisor functions on average

Abstract

Let $\ddot{I}{„}_k$ be the $k$-fold divisor function. By constructing an approximant of $\ddot{I}{„}_k$, denoted as $\ddot{I}{„}_k^{\ast }$, which is a normalized truncation of the $k$-fold divisor function, we prove that when $\exp \left(C{\log }^{1/2}X(\log \log X{)}^{1/2}\right)\le H\le X$ and $C>0$ is sufficiently large, the following estimate holds for almost all $x\in [X,2X]$: \[ \Big|\sum_{x<n\leq x+H}(τ_k(n)-τ_k^*(n)) e(α_dn^d+\cdots+α_1n)\Big|=o(H\log^{k-1}X), \] where $\hat{I}{\pm }_1,\dots ,\hat{I}{\pm }_d\in \mathbb{R}$ are arbitrary frequencies.

Version 2: corrected some minor typos