Kloosterman Sums with Multiplicative Coefficients

Abstract

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

In this version we make some refinement

Cited by in corpus (1)

• Shifted Character Sums with Multiplicative Coefficients