Sign changes of Kloosterman sums with moduli having at most six prime factors

TL;DR

This work addresses the sign changes of the Kloosterman sum $\text{Kl}(1;q)$ for square-free moduli with at most six prime factors. It introduces a new truncated divisor function $\tau(n;\alpha,\beta)$ whose subset selection depends on the number of prime factors, and fuses Selberg sieve weights with spectral and distributional methods for Kloosterman sums to analyze weighted sums $R^{\pm}(X)$. The authors prove that $\text{Kl}(1;q)$ changes sign infinitely often when $q$ is square-free with $\omega(q)\le 6$, improving the prior bound by Xi (IMRN 2022) within this framework. The approach combines a BDH-type bound for $|\text{Kl}|$, Mellin-transform techniques, contour integrals, and a careful combinatorial argument to constrain the prime-factor structure, offering a path toward further extensions with refined sieve and analytic tools.

Abstract

We prove that the Kloosterman sum $\text{Kl}(1,q)$ changes sign infinitely many times, as $q\rightarrow +\infty$ with at most six prime factors. As a consequence, our result improved the best known result of Xi(IMRN, 2022). The novelty of our method comes from introducing a new truncated divisor function whose selection depends on the number of prime factors of the variable, through which Kloosterman sum is controlled good enough. Our arguments contain the Selberg sieve method, spectral theory and distribution of Kloosterman sums along with previous nice works by Fouvry, Matomäki, Michel, Sivak-Fischler and Xi.

Theorems & Definitions (20)

• Conjecture 1.1
• Theorem 1.2
• Proposition 2.1
• Proposition 2.2
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof