Moment estimates for the exponential sum with higher divisor functions

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

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