Sign changes of Kloosterman sums with moduli having at most two 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 two prime factors. As a consequence, our result is unconditional compared with Drappeau and Maynard's (Proc. Amer. Math. Soc., 2019), in which the existence of Laudau-Siegel zeros is required. Our arguments contain the Selberg sieve method, spectral theory and distribution of Kloosterman sums along with previous works by Fouvry, Matom\"{a}ki, Michel, Sivak-Fischler, and Xi.