# Moment estimates for the exponential sum with higher divisor functions

## TL;DR

This paper derives asymptotic estimates for exponential sums involving higher divisor functions using the circle method, providing new bounds and moment asymptotics relevant to analytic number theory.

## Contribution

It introduces novel asymptotic formulas and minor arc bounds for exponential sums with higher divisor functions, advancing understanding of their behavior.

## Key findings

- Asymptotic formulas for integrals of exponential sums with τ_k
- Minor arc bounds for exponential sums with τ_k
- Asymptotics for high moments of the Dirichlet kernel

## Abstract

We obtain asymptotic for the quantity $\int_0^1 \bigg|\sum_{n\le X}\tau_k(n)e(n\alpha)\bigg|d\alpha$ where $\tau_k(n) = \sum_{d_1\dots d_k = n} 1$. This follows from a quick application of the circle method. Along the way, we find minor arc bounds for the exponential sum with $\tau_k$, and asymptotics for high moments of the Dirichlet kernel.