Mingxuan Zhong | Searcharxiv

arXiv2026

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

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.

math.NT↗

arXiv2025

A new result on the divisor problem in arithmetic progressions modulo a prime power

We derive an asymptotic formula for the divisor function $τ(k)$ in an arithmetic progression $k\equiv a(\bmod \ q)$, uniformly for $q\leq X^{Δ_{n,l}}$ with $(q,a)=1$. The parameter $Δ_{n,l}$ is defined as $$ Δ_{n,l}=\frac{1-\frac{3}{2^{2^l+2l-3}}}{1-\frac{1}{n2^{l-1}}}. $$ Specifically, by setting $l=2$, we achieve $Δ_{n,l}>3/4+5/32$, which surpasses the result obtained by Liu, Shparlinski, and Zhang (2018). Meanwhile, this has also improved upon the result of Wu and Xi (2021). Notably, Hooley, Linnik, and Selberg (1950's) independently established that the asymptotic formula holds for $q\leq X^{2/3-\varepsilon}$. Irving (2015) was the first to surpass the $2/3-$barrier for certain special moduli. We break the classical $3/4-$barrier in the case of prime power moduli and extend the range of $q$. Our main ingredients borrow from Mangerel's (2021) adaptation of Milićević and Zhang's methodology in dealing with a specific class of weighted Kloosterman sums, rather than adopting Korobov's technique employed by Liu, Shparlinski, and Zhang (2018).

math.NT↗