Kloosterman sums with multiplicative coefficients

Abstract

Let f(n) be a multiplicative function satisfying |f(n)| ≤ 1, q (≤ N ²) be a positive integer and a be an integer with (a, q) = 1. In this paper, we shall prove that

$\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)}},} $

where $\overline n$ is the multiplicative inverse of {itn} such that $\overline n n$ ≡ 1 (mod {itq}), {ite}({itx}) = exp(2πi{itx}), and τ(·) is the divisor function.