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On the Mean Value of $D_k(n)$ in Arithmetic Progressions

Meselem Karras

Abstract

Let $k \ge 2$ be a fixed integer. We define the multiplicative function $D_k(n) = d_k(n)/d_k^*(n)$, such that $d_k(n)$ is the Piltz divisor function and $d_k^*(n) = k^{ω(n)}$ is its unitary analogue, where $ω(n)$ is the number of distinct prime divisors of $n$. We establish an asymptotic formula for the sum \[ \sum_{\substack{n \le x \\ n \equiv a \pmod q}} D_k(n), \] where $\gcd(a,q)=1$. This result is a generalization of the study presented in \cite{Derbal 2023}. \noindent

On the Mean Value of $D_k(n)$ in Arithmetic Progressions

Abstract

Let be a fixed integer. We define the multiplicative function , such that is the Piltz divisor function and is its unitary analogue, where is the number of distinct prime divisors of . We establish an asymptotic formula for the sum where . This result is a generalization of the study presented in \cite{Derbal 2023}. \noindent
Paper Structure (4 sections, 8 theorems, 51 equations, 1 table)

This paper contains 4 sections, 8 theorems, 51 equations, 1 table.

Table of Contents

  1. Introduction
  2. A useful auxiliary theorem
  3. Main result
  4. Declaration of no conflict of interest

Key Result

Theorem 2.1

For any $x \geq 2$ and any $\varepsilon > 0$, we have where $\omega(q)$ denotes the number of distinct prime factors of $q$ and In particular, if $q = O(x^\delta)$ for some fixed $\delta > 0$, then for any $\varepsilon > 0$,

Theorems & Definitions (15)

  • Theorem 2.1
  • Lemma 2.2: Lemma 4 in Karras 2018
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • proof : Proof of Theorem \ref{['thm:1']}
  • Theorem 3.1
  • ...and 5 more