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<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(q_1\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(q_2\)</EquationSource> </InlineEquation> be two fixed positive integers, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\chi \)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\lambda \)</EquationSource> </InlineEquation> be the Dirichlet characters modulo <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(q_1\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(q_2\)</EquationSource> </InlineEquation>, respectively. We obtain asymptotic formulas for <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(\sum _{mn\le x}f\big (\gcd (m, n)\big )\chi (m) \lambda (n)\)</EquationSource> </InlineEquation>, where <i>f</i> belongs to a certain classes of arithmetic functions and <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(\gcd (m, n)\)</EquationSource> </InlineEquation> denoting the greatest common divisor of the integers <i>m</i>,&#xa0;<i>n</i>. An application on the distribution of its value in an arithmetic progression is given in this paper.</p>

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On certain sums of arithmetic functions involving the greatest common divisor and its applications

  • 期刊论文

  • 发表日期：

    2026年4月28日

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  • Wenpeng Zhang,
  • Teerapat Srichan

摘要

Let \(q_1\) and \(q_2\) be two fixed positive integers, \(\chi \) and \(\lambda \) be the Dirichlet characters modulo \(q_1\) and \(q_2\) , respectively. We obtain asymptotic formulas for \(\sum _{mn\le x}f\big (\gcd (m, n)\big )\chi (m) \lambda (n)\) , where f belongs to a certain classes of arithmetic functions and \(\gcd (m, n)\) denoting the greatest common divisor of the integers m, n. An application on the distribution of its value in an arithmetic progression is given in this paper.