Kloosterman sums with multiplicative coefficients

Let f(n) be a multiplicative function satisfying |f(n)| ≤ 1, q (≤ N2) be a positive integer and a be an integer with (a, q) = 1. In this paper, we shall prove that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\sum\limits_{n \leqslant N(n,q) = 1} {f(n)e\left( {\frac{{a\bar n}} {q}} \right) \ll \sqrt {\frac{{\tau (q)}} {q}} N\log \log (6N) + q\tfrac{1} {4} + \tfrac{\varepsilon } {2}N\tfrac{1} {2}(\log (6N))\tfrac{1} {2} + \frac{N} {{\log \log (6N)}},} $\end{document} where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\overline n $\end{document} is the multiplicative inverse of {itn} such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\overline n n$\end{document} ≡ 1 (mod {itq}), {ite}({itx}) = exp(2πi{itx}), and τ(·) is the divisor function.