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<p>Let <i>x</i> &gt; 1 and <i>α, β</i> be positive integers such that 1 &lt; <i>α</i> &lt; <i>β</i>. We consider sums of type <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\sum_{m^\alpha n^\beta\leq x}f(\gcd(m, n)\!)\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <munder> <mo>∑</mo> <mrow> <msup> <mi>m</mi> <mi>α</mi> </msup> <msup> <mi>n</mi> <mi>β</mi> </msup> <mo>≤</mo> <mi>x</mi> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mo form="prefix" movablelimits="true">gcd</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mspace width="negativethinmathspace" /> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation>, taken over the region {(<i>m,n</i>) ∈ ℕ<sup>2</sup>: <i>m</i><sup><i>α</i></sup><i>n</i><sup><i>β</i></sup> ⩽ <i>x</i>}, where <i>f</i> belongs to certain classes of arithmetic functions and gcd(<i>m, n</i>) denotes the greatest common divisor of the integers <i>m, n</i>.</p>

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On certain sums of arithmetic functions involving the greatest common divisor

  • 期刊论文

  • 发表日期：

    2026年2月10日

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  • Traiwat Intarawong,
  • Teerapat Srichan,
  • Boonrod Yuttanan

摘要

Let x > 1 and α, β be positive integers such that 1 < α < β. We consider sums of type \(\sum_{m^\alpha n^\beta\leq x}f(\gcd(m, n)\!)\) ∑ m α n β ≤ x f ( gcd ( m , n ) ) , taken over the region {(m,n) ∈ ℕ2: mαnβ ⩽ x}, where f belongs to certain classes of arithmetic functions and gcd(m, n) denotes the greatest common divisor of the integers m, n.