Kloosterman Sums with Multiplicative Coefficients

Let $f(n)$ be a multiplicative function satisfying $\mathrm{\mid }f(n)\mathrm{\mid }\le 1$, $q$ $(\leq N^2)$beapositiveintegerand$a$beanintegerwith$(a,,q)=1$. In this paper, we shall prove that $$\sum _{\begin{array}{c}n\le N\\ (n,q)=1\end{array}}f(n)e(\frac{a\bar{n}}{q})\ll \sqrt{\frac{\tau (q)}{q}}N\log \log (6N)+q^{\frac{1}{4}+\frac{ϵ}{2}}{N}^{\frac{1}{2}}(\log (6N){)}^{\frac{1}{2}}+\frac{N}{\sqrt{\log \log (6N)}},$$ where $\bar{n}$ is the multiplicative inverse of $n$ such that $\bar{n}n\equiv 1,({\rm mod},q),,e(x)=\exp(2\pi ix),,\tau(q)$ is the divisor function.