Exponential sums involving the divisor function over arithmetic progressions

Abstract

Let $ \phi(x) $ be a smooth function supported on $ [1, 2] $ with derivatives bounded by $ \phi^{(j)}(x)\ll 1 $ and $ d_3(n) $ be the number of ways to write $ n $ as a product of three factors. We get the asymptotic formula for the nonlinear exponential sum $ \sum\limits_{n\ \equiv\ l\ mod\ q}d_3(n)\phi\left(\frac{n}{X}\right)e\left(\frac{3\sqrt[3]{kn}}{q}\right) $.