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The summatory function of the Möbius function involving the greatest common divisor

Isao Kiuchi, Sumaia Saad Eddin

Abstract

Let $\gcd(m,n)$ denote the greatest common divisor of the positive integers $m$ and $n$, and let $μ$ represent the M\" obius function. For any real number $x>5$, we define the summatory function of the M\" obius function involving the greatest common divisor as $ S_μ(x) := \sum_{mn\leq x} μ(\gcd(m,n)). $ In this paper, we present an asymptotic formula for $S_μ(x)$. Assuming the Riemann Hypothesis, we delve further into the asymptotic behavior of $S_μ(x)$ and derive a mean square estimate for its error term. Our proof employs the Perron formula, Parseval's theorem, complex integration techniques, and the properties of the Riemann zeta-function.

The summatory function of the Möbius function involving the greatest common divisor

Abstract

Let denote the greatest common divisor of the positive integers and , and let represent the M\" obius function. For any real number , we define the summatory function of the M\" obius function involving the greatest common divisor as In this paper, we present an asymptotic formula for . Assuming the Riemann Hypothesis, we delve further into the asymptotic behavior of and derive a mean square estimate for its error term. Our proof employs the Perron formula, Parseval's theorem, complex integration techniques, and the properties of the Riemann zeta-function.
Paper Structure (8 sections, 12 theorems, 75 equations)

This paper contains 8 sections, 12 theorems, 75 equations.

Table of Contents

  1. Introduction and main results
  2. Some known results
  3. Proofs
  4. Proof of Theorem \ref{['th1']}
  5. Proof of Theorem \ref{['th31']}
  6. Proof of Theorem \ref{['th4']}
  7. Proof of Corollary \ref{['cor1']}
  8. Proof of Theorem \ref{['th5']}

Key Result

Theorem 1

For any real number $x>5$, we have where $\gamma$ is the Euler-Mascheroni constant and $\zeta^{\prime}(s)$ is the first derivative of the zeta-function $\zeta(s).$

Theorems & Definitions (15)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Corollary 1
  • Theorem 4
  • Remark 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • ...and 5 more