Distinction between hyper-Kloosterman sums and multiplicative functions
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| Format: | Preprint |
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2025
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| _version_ | 1866915667360350208 |
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| author | Zhang, Yang |
| author_facet | Zhang, Yang |
| contents | Let $\kl_n(a,b;m)$ be the hyper-Kloosterman sum. Fix integers $n\geqslant2,a\neq0$, $b\neq0$ and $k\geqslant2$. For any $0\neqη\in\mathbb{C}$ and multiplicative function $f: \mathbb{N} \rightarrow \mathbb{C}$, 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_10721 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Distinction between hyper-Kloosterman sums and multiplicative functions Zhang, Yang Number Theory 11L05, 11N64, 11N37, 11T23 Let $\kl_n(a,b;m)$ be the hyper-Kloosterman sum. Fix integers $n\geqslant2,a\neq0$, $b\neq0$ and $k\geqslant2$. For any $0\neqη\in\mathbb{C}$ and multiplicative function $f: \mathbb{N} \rightarrow \mathbb{C}$, 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. |
| title | Distinction between hyper-Kloosterman sums and multiplicative functions |
| topic | Number Theory 11L05, 11N64, 11N37, 11T23 |
| url | https://arxiv.org/abs/2510.10721 |