Kloosterman Sums with Multiplicative Coefficients

TL;DR

This paper establishes an upper bound for sums involving multiplicative functions and Kloosterman sums with multiplicative coefficients, extending understanding of exponential sums in number theory.

Contribution

It provides a new upper bound for Kloosterman sums with multiplicative coefficients, combining techniques from multiplicative number theory and exponential sum estimates.

Findings

Derived an explicit upper bound for the sum involving f(n) and exponential terms.

Extended classical bounds to sums with multiplicative coefficients and Kloosterman sums.

Enhanced understanding of the distribution of multiplicative functions in exponential sums.

Abstract

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 $$\sum_{\substack{n\leq N\\ (n,\,q)=1}}f(n)e({a\bar{n}\over q})\ll\sqrt{\tau(q)\over q}N\log\log(6N)+q^{{1\over 4}+{\epsilon\over 2}}N^{1\over 2}(\log(6N))^{1\over 2}+{N\over \sqrt{\log\log(6N)}},$$ where $\bar{n}$ is the multiplicative inverse of $n$ such that $\bar{n}n\equiv 1\,({\rm mod}\,q),\,e(x)=\exp(2\pi ix),\,\tau(q)$ is the divisor function.