On the distinction between Kloosterman sums and multiplicative functions

Abstract: Let $S(a,b;n)$ be the Kloosterman sum. For any integers $a\neq0,b\neq0,k\geqslant2$, any given complex number $\eta\not=0$ and multiplicative function $f: \mathbb{N} \rightarrow \mathbb{C}$, we show that [ |{n\leqslant X:S(a,b;n)=\eta f(n), n \text{ square-free }k\text{-almost prime}}|\leqslant \pi(\dfrac{X}{L_k})+O(X^{1-\frac{1}{2k}}) ] and [ |{n\leqslant X:S(a,b;n)=\eta f(n), n \text{ square-free}}|\leqslant \pi(X)+O(Xe^{-\sqrt{\log X}}) ] as $X\rightarrow \infty$. Here, $L_{k}$ denotes the product of the first $(k-1)$ primes and $\pi(X)$ is the number of primes $\leqslant X$. By providing some examples for which the equality holds, we also prove that these two inequalities are sharp. In particular, these estimates imply that $S(a,b;n)\neq\eta f(n)$ holds for $100\%$ square-free $k$-almost prime numbers and $100\%$ square-free numbers $n$. Counterintuitively, if $S(a,b;p)=\eta f(p)$ holds for all but finitely many primes $p$, we further show that \begin{align*} |{n\leqslant X:S(a,b;n)=\eta f(n), n \text{ square-free }k\text{-almost prime}}|\leqslant (|a|+|b|+2k)X^{\frac{k}{k+1}}+O(1). \end{align*} All these demonstrate that $S(a,b;n)$ with fixed $a\neq0$ and $b\neq0$, as an arithmetic function of $n\in\mathbb{N}$ is intrinsically distinct from any complex valued multiplicative function. Similarly, we also show that all these results still hold with the Birch sum $B(a,b;n)$ or the Sali\'e sum $\widetilde{S}(a,b;n)$ in place of $S(a,b;n)$ if and only if the integer pair $a,b$ satisfies some natural conditions respectively.