Year: 2025
Type: article
Abstract: 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).
Source: Scientia Sinica Mathematica
Authors: Zhong Mingxuan, Zhang Tianping
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G3628969712, 指数和、特征和的相消性及相关问题研究
OpenAlex ID: W4415200246
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