Distinction between hyper-Kloosterman sums and multiplicative functions

Abstract

Let $\kl_n(a,b;m)$ be the hyper-Kloosterman sum. Fix integers $n⩾2,a\ne 0$, $b\ne 0$ and $k⩾2$. For any $0\ne \eta \in \mathbb{C}$ and multiplicative function $f:\mathbb{N}\to \mathbb{C}$, we prove that $\kl_n(a,b;m)\neqηf(m)$ holds for $100\%$ square-free $k$-almost prime numbers $m$ and $100\%$ square-free numbers $m$. Counterintuitively, if $\kl_n(a,b;p)=ηf(p)$ holds for all but finitely many primes $p$, we further show that \begin{align*} \ab|\{m\leqslant X:\kl_n(a,b;m)=ηf(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.

We have extended all the main results in the previous version to hyper-kloosterman sums, and the title of the paper adjusted accordingly