Moment estimates for the exponential sum with higher divisor functions

Abstract

We obtain asymptotic for the quantity ${\int }_0^1|{\sum }_{n\le X}{\tau }_k(n)e(n\alpha )|d\alpha$ 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.

6 pages; Incorporated corrections following referee report. Strength of error term corrected, and minor arc bound strengthened