Distinction between hyper-Kloosterman sums and multiplicative functions

Abstract

Let $\kl_n(a,b;m)$ be the hyper-Kloosterman sum. Fix integers $n\geqslant2,a\neq0$, $b\neq0$ and $k\geqslant2$. For any $0\neq\eta\in\mathbb{C}$ and multiplicative function $f: \mathbb{N} \rightarrow \mathbb{C}$, we prove that $\kl_n(a,b;m)\neq\eta f(m)$ holds for $100\%$ square-free $k$-almost prime numbers $m$ and $100\%$ square-free numbers $m$. Counterintuitively, if $\kl_n(a,b;p)=\eta f(p)$ holds for all but finitely many primes $p$, we further show that \begin{align*} \ab|\{m\leqslant X:\kl_n(a,b;m)=\eta f(m), m \text{ square-free }k\text{-almost prime}\}|= O(X^{1-\frac{1}{k+1}}). \end{align*} These results overturn the general belief that $\kl_n(a,b;m)$ is nearly multiplicative in $m$, and that its distribution at almost prime moduli $m$ closely approximates that at primes. Moreover, we prove that these results also hold for general algebraic exponential sums satisfying some natural conditions.