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

TL;DR

This paper derives an improved asymptotic formula for the divisor function in arithmetic progressions modulo prime powers, extending the range of q and surpassing classical barriers using novel methods involving weighted Kloosterman sums.

Contribution

It introduces a new approach to analyze the divisor function in prime power moduli, surpassing the classical 3/4 barrier and extending the range of q with innovative techniques.

Findings

Achieved asymptotic formula for divisor function in prime power moduli.

Extended the range of q beyond the classical 3/4 barrier.

Introduced a new method involving weighted Kloosterman sums.

Abstract

We derive an asymptotic formula for the divisor function $\tau(k)$ in an arithmetic progression $k\equiv a(\bmod \ q)$, uniformly for $q\leq X^{\Delta_{n,l}}$ with $(q,a)=1$. The parameter $\Delta_{n,l}$ is defined as $$ \Delta_{n,l}=\frac{1-\frac{3}{2^{2^l+2l-3}}}{1-\frac{1}{n2^{l-1}}}. $$ Specifically, by setting $l=2$, we achieve $\Delta_{n,l}>3/4+5/32$, 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 Mili\'{c}evi\'{c} 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).