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

Mingxuan Zhong, Tianping Zhang

TL;DR

This work advances the divisor problem in arithmetic progressions modulo prime-power moduli by establishing an asymptotic formula for the divisor function in the progression k \equiv a (mod q) uniformly for q up to X^{Δ_{n,l}} for certain prime powers q=p^{n}. The authors define Δ_{n,l} = \frac{1-\frac{3}{2^{2^l+2l-3}}}{1-\frac{1}{n2^{l-1}}} and prove that, for l\ge 2 and n>2^{2^{l+1}+6l}, the bound max_{(q,a)=1}|E(q,a;X)| \prec\prec_{p,l} X^{1-\varepsilon}/q holds uniformly when q \le X^{Δ_{n,l}}, with Δ_{n,l} exceeding 29/32 in the case l=2. The key methodological shift is to leverage Milicevic–Zhang-type weighted Kloosterman-sum bounds (rather than Korobov’s approach) to control exponential-sum structures arising from the modulus p^{n}, thereby pushing the q-range beyond previously known barriers and connecting the mean-value analysis to a refined analysis of weighted Kloosterman sums. The results extend the reach of divisor-problem asymptotics to prime-power moduli and highlight a trade-off between the admissible modulus range and the complexity of the exponential-sum bounds required. Overall, the paper couples dyadic decompositions, p-adic stationary-phase techniques, and sophisticated exponential-sum bounds to surpass classical barriers and broaden the applicability of asymptotic divisor formulas in arithmetic progressions.

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).

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

TL;DR

This work advances the divisor problem in arithmetic progressions modulo prime-power moduli by establishing an asymptotic formula for the divisor function in the progression k \equiv a (mod q) uniformly for q up to X^{Δ_{n,l}} for certain prime powers q=p^{n}. The authors define Δ_{n,l} = \frac{1-\frac{3}{2^{2^l+2l-3}}}{1-\frac{1}{n2^{l-1}}} and prove that, for l\ge 2 and n>2^{2^{l+1}+6l}, the bound max_{(q,a)=1}|E(q,a;X)| \prec\prec_{p,l} X^{1-\varepsilon}/q holds uniformly when q \le X^{Δ_{n,l}}, with Δ_{n,l} exceeding 29/32 in the case l=2. The key methodological shift is to leverage Milicevic–Zhang-type weighted Kloosterman-sum bounds (rather than Korobov’s approach) to control exponential-sum structures arising from the modulus p^{n}, thereby pushing the q-range beyond previously known barriers and connecting the mean-value analysis to a refined analysis of weighted Kloosterman sums. The results extend the reach of divisor-problem asymptotics to prime-power moduli and highlight a trade-off between the admissible modulus range and the complexity of the exponential-sum bounds required. Overall, the paper couples dyadic decompositions, p-adic stationary-phase techniques, and sophisticated exponential-sum bounds to surpass classical barriers and broaden the applicability of asymptotic divisor formulas in arithmetic progressions.

Abstract

We derive an asymptotic formula for the divisor function in an arithmetic progression , uniformly for with . The parameter is defined as Specifically, by setting , we achieve , 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 . Irving (2015) was the first to surpass the barrier for certain special moduli. We break the classical barrier in the case of prime power moduli and extend the range of . 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).
Paper Structure (9 sections, 10 theorems, 111 equations)

This paper contains 9 sections, 10 theorems, 111 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Our results
  4. Kloosterman sums
  5. Reduction of the problem
  6. Proof of Theorem \ref{['thm: main theorem']}
  7. Proof of Theorem \ref{['thm: weighted Kloosterman sums bound']}
  8. Preliminary lemmas
  9. Upper bound for $S(N)$

Key Result

Theorem 1.1

Let $l\ge 2$, $n>2^{2^{l+1}+6l}$ be fixed integers. Let $q=p^n$ and $a$ be integers, with $p$ an odd prime and $(q,a)=1$. For any sufficiently small constant $\varepsilon>0$, there exists a constant $\Delta_{n,l}$ that depends only on $l$ and $n$, such that holds uniformly for $q\leq X^{\Delta_{n,l}}$. Specifically, we have

Theorems & Definitions (17)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 5.1
  • proof
  • Lemma 5.2
  • ...and 7 more