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

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.