Moment estimates for the exponential sum with higher divisor functions (Q2143608)

Let \(k\geq 2\) be some integer, \(s>2\) be real, and \(\tau_k(n)\) be the generalized divisor counting function. In this paper, the author provides asymptotic expansion for the moments \[ \int_0^1\left|\sum_{n\leq X}\tau_k(n)e^{2\pi i\alpha n}\right|^s\,d\alpha. \] To deal with the main term, the author applies the following approximation on the order of magnitude of higher moments of Dirichlet kernels, \[ \int_0^1\left|\sum_{n\leq X}e^{2\pi i\alpha n}\right|^s\,d\alpha=\left(\frac{2}{\pi}\int_0^\infty\frac{|\sin t|^s}{t^s}\,dt\right)X^{s-1}+O_s(X^{s-2}). \]