On Kloosterman sums with multiplicative coefficients

TL;DR

This paper develops new estimates for sums involving multiplicative functions and Kloosterman sums, extending understanding of their behavior for large moduli and specific ranges of summation.

Contribution

It introduces novel bounds for sums of multiplicative functions combined with Kloosterman sums, expanding previous results in analytic number theory.

Findings

Derived new estimates for sums with multiplicative coefficients and Kloosterman sums

Extended the range of x for which bounds are valid

Improved understanding of the distribution of such sums in analytic number theory

Abstract

The series of some new estimates for the sums of the type \[ S_{q}(x;f)\,=\,\mathop{{\sum}'}\limits_{n\leqslant x}f(n)e_{q}(an^{*}+bn) \] is obtained. Here $q$ is a sufficiently large integer, $\sqrt{q}(\log{q})\!\ll\!x\leqslant q$, $a,b$ are integers, $(a,q)=1$, $e_{q}(v) = e^{2\pi iv/q}$, $f(n)$ is a multiplicative function, $nn^{*}\equiv 1 \pmod{q}$ and the prime sign means that $(n,q)=1$.