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) $.
Citation: Rui Zhang, Yang Li, Xiaofei Yan. Exponential sums involving the divisor function over arithmetic progressions[J]. AIMS Mathematics, 2023, 8(5): 11084-11094. doi: 10.3934/math.2023561
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) $.
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Rui Zhang, Yang Li, Xiaofei Yan. Exponential sums involving the divisor function over arithmetic progressions[J]. AIMS Mathematics, 2023, 8(5): 11084-11094. doi: 10.3934/math.2023561
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