Kloosterman Sums with Multiplicative Coefficients

Let $f(n)$ be a multiplicative function satisfying $|f(n)|\leq 1$, $q$ $(\leq N^2)$ be a positive integer and $a$ be an integer with $(a,\,q)=1$. In this paper, we shall prove that $\sum_{\substack{n\leq N\\ (n,\,q)=1}}f(n)e({a\bar{n}\over q})\ll\sqrt{τ(q)\over q}N\log\log(6N)+q^{{1\over 4}+{ε\over 2}}N^{1\over 2}(\log(6N))^{1\over 2}+{N\over \sqrt{\log\log(6N)}},$ where $\bar{n}$ is the multiplicative inverse of $n$ such that $\bar{n}n\equiv 1\,({\rm mod}\,q),\,e(x)=\exp(2πix),\,τ(q)$ is the divisor function.

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