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On the Mean Value of a Weighted Composite Arithmetic Function

Mihoub Bouderbala

Abstract

The primary objective of this paper is to employ methods from analytic number theory to investigate the mean value properties of a composite function involving the Dirichlet divisor function and a generalized minimal power function. Specifically, we study the weighted summatory function where the divisor function is normalized by the number of distinct prime factors. We establish a rigorous asymptotic formula for this sum, detailing the analytic properties of the associated Dirichlet series and the contour integration process.

On the Mean Value of a Weighted Composite Arithmetic Function

Abstract

The primary objective of this paper is to employ methods from analytic number theory to investigate the mean value properties of a composite function involving the Dirichlet divisor function and a generalized minimal power function. Specifically, we study the weighted summatory function where the divisor function is normalized by the number of distinct prime factors. We establish a rigorous asymptotic formula for this sum, detailing the analytic properties of the associated Dirichlet series and the contour integration process.
Paper Structure (4 sections, 5 theorems, 45 equations)

This paper contains 4 sections, 5 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Preliminary Lemmas
  3. Proof of the Main Result
  4. Conclusion

Key Result

Theorem 1

Let $r \ge 2$ be a fixed integer and $k \ge 1$ be a real parameter. For any real number $x \ge 1$, the following asymptotic formula holds: where and such that,

Theorems & Definitions (10)

  • Theorem 1
  • Remark 2
  • Lemma 3: Euler Product Representation
  • proof
  • Lemma 4: Perron's Formula
  • proof
  • Lemma 5: Bounds for $\zeta(s)$
  • proof
  • Lemma 6: Summation of Arithmetic Progressions in Exponents
  • proof