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<p>For a natural number <i>n</i>, let <i>M</i>(<i>n</i>) denote the maximum exponent of any prime power dividing <i>n</i>, and let <i>m</i>(<i>n</i>) denote the minimum exponent of any prime power dividing <i>n</i>. We study the second moments of these arithmetic functions and establish their limiting distributions. We introduce a new discrete probabilistic distribution dependent on a function <i>f</i> taking values in [0,&#xa0;1], study its first two moments, and provide examples of several arithmetic functions satisfying such a distribution as their limiting behavior.</p>

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On limiting distributions of arithmetic functions

  • 期刊论文

  • 发表日期：

    2025年9月1日

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  • Sourabhashis Das

摘要

For a natural number n, let M(n) denote the maximum exponent of any prime power dividing n, and let m(n) denote the minimum exponent of any prime power dividing n. We study the second moments of these arithmetic functions and establish their limiting distributions. We introduce a new discrete probabilistic distribution dependent on a function f taking values in [0, 1], study its first two moments, and provide examples of several arithmetic functions satisfying such a distribution as their limiting behavior.