Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\gcd (m,n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo movablelimits="true">gcd</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> denote the greatest common divisor of the positive integers m and n, and let <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation> be the Möbius function. For any real number <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$(x > 5)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>></mo> <mn>5</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, define the summatory function involving the greatest common divisor by <Equation ID="Equ38"> <EquationSource Format="TEX">$ S_{\mu }(x) := \sum _{mn \le x} \mu (\gcd (m,n)). $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msub> <mi>S</mi> <mi>μ</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>:</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow> <mi>m</mi> <mi>n</mi> <mo>≤</mo> <mi>x</mi> </mrow> </munder> <mi>μ</mi> <mrow> <mo stretchy="false">(</mo> <mo movablelimits="true">gcd</mo> <mrow> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mrow> </math></EquationSource> </Equation>In this paper, we establish an asymptotic formula for <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$S_{\mu }(x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>S</mi> <mi>μ</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Under the assumption of the Riemann Hypothesis, we further refine this formula and derive a mean square estimate for the corresponding error term.

On Sums of the Möbius Function Involving the Greatest Common Divisor

期刊论文
发表日期：

2026年3月30日

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Isao Kiuchi,
Sumaia Saad Eddin

摘要

Let $\gcd (m,n)$ $gcd (m, n)$ denote the greatest common divisor of the positive integers m and n, and let $\mu $ $μ$ be the Möbius function. For any real number $(x > 5)$ $(x > 5)$ , define the summatory function involving the greatest common divisor by $ S_{\mu }(x) := \sum _{mn \le x} \mu (\gcd (m,n)). $ $S_{μ} (x) : = \sum_{m n \leq x} μ (gcd (m, n)) .$ In this paper, we establish an asymptotic formula for $S_{\mu }(x)$ $S_{μ} (x)$ . Under the assumption of the Riemann Hypothesis, we further refine this formula and derive a mean square estimate for the corresponding error term.