Distinction between hyper-Kloosterman sums and multiplicative functions

Yang Zhang

Paper

Distinction between hyper-Kloosterman sums and multiplicative functions

Abstract

Let

be the hyper-Kloosterman sum. Fix integers

and

. For any

and multiplicative function

, we prove that

holds for

square-free

-almost prime numbers

and

square-free numbers

. Counterintuitively, if

holds for all but finitely many primes

, 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

is nearly multiplicative in

, and that its distribution at almost prime moduli

closely approximates that at primes. Moreover, we prove that these results also hold for general algebraic exponential sums satisfying some natural conditions.