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

TL;DR

This paper proves that Kloosterman sums with moduli having up to six prime factors change sign infinitely often, improving previous results by employing a novel divisor function, sieve methods, and spectral theory.

Contribution

Introduces a new truncated divisor function depending on the number of prime factors, enabling control of Kloosterman sums and proving their sign changes infinitely often.

Findings

Kloosterman sums change sign infinitely often for moduli with up to six prime factors.

The method improves upon previous results by Xi (2022).

Uses a combination of sieve, spectral theory, and distribution analysis.

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, Matom\"{a}ki, Michel, Sivak-Fischler and Xi.