| Abstract: |
| Many problems in number theory are highly related to the bound of the following exponential sum
$$
\sum_{h \sim H} \bigg| \sum_{n \sim x} \Lambda(n) \e(h n^\gamma + \alpha n) \bigg| \ll x^{1-\epsilon}
$$
We give a new estimation of the type I bound to improve the admissible range of $\gamma$ to $\frac{580}{663} < \gamma |
|