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<p>Let <i>K</i> be a quadratic field, <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathcal {O}_K\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">O</mi> <mi>K</mi> </msub> </math></EquationSource> </InlineEquation> its ring of integers, and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathfrak {a}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="fraktur">a</mi> </math></EquationSource> </InlineEquation> a square-free ideal of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\mathcal {O}_K\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">O</mi> <mi>K</mi> </msub> </math></EquationSource> </InlineEquation>. In this paper, we investigate the distribution of ideals that divide <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\mathfrak {a}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="fraktur">a</mi> </math></EquationSource> </InlineEquation> using the Selberg–Delange method. We established that the Cesàro mean of distribution functions converges uniformly to the arcsine law in short intervals. This generalizes the result studied by Deshouillers–Dress–Tenenbaum and Cui–Wu.</p>

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The arcsine law on divisors of square-free ideals in short intervals

  • 期刊论文

  • 发表日期：

    2026年7月24日

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  • Tingting Wen

摘要

Let K be a quadratic field, \(\mathcal {O}_K\) O K its ring of integers, and \(\mathfrak {a}\) a a square-free ideal of \(\mathcal {O}_K\) O K . In this paper, we investigate the distribution of ideals that divide \(\mathfrak {a}\) a using the Selberg–Delange method. We established that the Cesàro mean of distribution functions converges uniformly to the arcsine law in short intervals. This generalizes the result studied by Deshouillers–Dress–Tenenbaum and Cui–Wu.