Distinction between hyper-Kloosterman sums and multiplicative functions

2025/10/12 by Zhang, Yang
Computer Science · Mathematics · #11L05 #11N37 #11N64 #11T23 #Advanced Mathematical Identities #Analytic Number Theory Research #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2510.10721

openalex publication_date 2025/10/12 · openalex created_date 2025/10/17 · openalex updated_date 2026/08/01

Abstract

Let \kl_n(a,b;m) be the hyper-Kloosterman sum. Fix integers n\geqslant2,a≠0, b≠0 and k\geqslant2. For any 0≠η∈ℂ and multiplicative function f: ℕ → ℂ, we prove that \kl_n(a,b;m)≠ηf(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 \ab|\m\leqslant X:\kl_n(a,b;m)=ηf(m), m square-free k-almost prime\|= O(X^1-(1)/(k+1)). 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.

Citations

• Sign changes of Kloosterman sums with almost prime moduli

Related

• Kloosterman Sums with Multiplicative Coefficients
• Multiplicative functions supported on the k-free integers with small partial sums
• On the distribution of Kloosterman sums
• Kloosterman sums with primes to composite moduli
• Sign changes of Kloosterman sums with almost prime moduli