%0 Journal Article
%T 高斯域上一种加权形式的Erd&#337;s-Kac定理<br>A Weighted Form of the Erd&#337;s-Kac Theorem over Gaussian Fields
%A 于宗祺
%J Pure Mathematics
%P 138-150
%@ 2160-7605
%D 2025
%I Hans Publishing
%R 10.12677/pm.2025.154117
%X Erd&#337;s-Kac定理是数论中的一个经典结果&#65292;它描述了在自然数范围内&#65292;整数的不同素因子个数的分布渐进服从正态分布。本文主要目的是将Erd&#337;s-Kac定理在高斯域中进行推广&#65292;令<math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mi>K</mi>
</math>

是高斯域&#65292;<math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <msub>
   <mi>O</mi>
   <mi>K</mi>
  </msub>
  </mrow>
</math>

是其整数环。设<math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mi>a</mi><mo>&#x2208;</mo><msub>
   <mi>O</mi>
   <mi>K</mi>
  </msub>
  </mrow>
</math>

&#65292;<math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mi>&#x03C9;</mi><mrow><mo>(</mo>
   <mi>a</mi>
  <mo>)</mo></mrow></mrow>
</math>

表示其不同的素因子个数&#65292;<math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <msub>
   <mi>&#x03C4;</mi>
   <mi>k</mi>
  </msub>
  <mrow><mo>(</mo>
   <mi>a</mi>
  <mo>)</mo></mrow></mrow>
</math>

是高斯域上<math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mi>k</mi>
</math>

重除数函数。我们用围道积分法&#65292;推导出<math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mi>&#x03C9;</mi><mrow><mo>(</mo>
   <mi>a</mi>
  <mo>)</mo></mrow></mrow>
</math>

的加权均值和<math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mi>m</mi>
</math>

阶中心矩&#65292;并由此推导出高斯域上权重为<math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <msub>
   <mi>&#x03C4;</mi>
   <mi>k</mi>
  </msub>
  <mrow><mo>(</mo>
   <mi>a</mi>
  <mo>)</mo></mrow></mrow>
</math>

的Erd&#337;s-Kac定理。这一结果不仅丰富了数论中的分布理论&#65292;也为进一步研究高斯域中的数论问题提供了新的工具和方法。<br>The Erd&#337;s-Kac theorem is a classical result in number theory, which describes that the distribution of the number of distinct prime factors of integers asymptotically follows a normal distribution. The primary aim of this paper is to extend the Erd&#337;s-Kac theorem to Gaussian fields. Let <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mi>K</mi>
</math>

 be a Gaussian field and <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <msub>
   <mi>O</mi>
   <mi>K</mi>
  </msub>
  </mrow>
</math>

be its ring of integers. Let <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mi>a</mi><mo>&#x2208;</mo><msub>
   <mi>O</mi>
   <mi>K</mi>
  </msub>
  </mrow>
</math>

, and <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mi>&#x03C9;</mi><mrow><mo>(</mo>
   <mi>a</mi>
  <mo>)</mo></mrow></mrow>
</math>

 denote the number of distinct prime factors of <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mi>a</mi>
</math>

. Let <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <msub>
   <mi>&#x03C4;</mi>
   <mi>k</mi>
  </msub>
  <mrow><mo>(</mo>
   <mi>a</mi>
  <mo>)</mo></mrow></mrow>
</math>

 be the
%K 除数函数&#65292
%K Erd&#337
%K s-Kac定理&#65292
%K 高斯域&#65292
%K 围道积分法<br>Divisor Function
%K Erd&#337
%K s-Kac Theorem
%K Gaussian Field
%K Contour Integration Method
%U http://www.hanspub.org/journal/PaperInformation.aspx?PaperID=111943