"정수에서의 리만제타함수의 값"의 두 판 사이의 차이

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<math>\sum_{n=1}^{\infty}\frac{1}{n^{4}}</math> is the [[Riemann zeta function#Values at the integers|Riemann zeta function]] evaluated for the argument 4, which is given by <math>\pi^{4}/90</math>. (See "Finding Zeta(4)" at [[Wallis product]] for a simple though lengthy derivation of <math>\zeta(4)</math>. This fact can also be proven by considering the following [[Methods of contour integration|contour integral]].) :<math>\oint_{C_{R}}\frac{\pi\cot(\pi z)}{z^{4}}.</math> Where <math>C_{R}</math> is a contour of radius <math>R</math> around the origin. In the limit, as <math>R</math> approaches infinity, the integral approaches zero. Using the [[residue theorem]] the integral can also be written as a sum of residues at the poles of the integrand. The poles are at zero, the positive and negative integers. The sum of the residues yields precisely twice the desired summation plus the residue at zero. Because the integral approaches zero, the sum of all the residues must be zero. The summation must therefore equal minus one half times the residue at zero. From the [[Trigonometric function#Series definitions|series expansion of the cotangent function]] :<math> \cot(x)=\frac{1}{x} - \frac {x}{3} - \frac {x^3} {45} +\ldots, </math> we see that the residue at zero is <math>-\pi^{4}/45</math> which yields the desired result.
  
 
 
 
 
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* [[ζ(2)의 계산, 오일러와 바젤문제(완전제곱수의 역수들의 합)|오일러와 바젤문제(완전제곱수의 역수들의 합)]]<br>
 
* [[모든 자연수의 곱과 리만제타함수]]<br>
 
* [[모든 자연수의 곱과 리만제타함수]]<br>
 
* [[모든 자연수의 합과 리만제타함수]]<br>
 
* [[모든 자연수의 합과 리만제타함수]]<br>
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* [[베르누이 수|베르누이 수와 베르누이 다항식]]<br>
  
 
 
 
 

2009년 7월 11일 (토) 14:19 판

간단한 소개
  • 홀수인 자연수를 제외한 모든 정수에 대하여 리만제타함수의 값은 닫힌 형태로 알려져 있음.
    \(\zeta(2n) =(-1)^{n+1}\frac{B_{2n}(2\pi)^{2n}}{2(2n)!}, n \ge 1\)여기서 \(B_{2n}\)은 베르누이수. 
    \(\zeta(-n)=-\frac{B_{n+1}}{n+1}, n \ge 1\)
    \(\zeta(0)=-\frac{1}{2}\)

 

 

증명

\(\sum_{n=1}^{\infty}\frac{1}{n^{4}}\) is the Riemann zeta function evaluated for the argument 4, which is given by \(\pi^{4}/90\). (See "Finding Zeta(4)" at Wallis product for a simple though lengthy derivation of \(\zeta(4)\). This fact can also be proven by considering the following contour integral.) \[\oint_{C_{R}}\frac{\pi\cot(\pi z)}{z^{4}}.\] Where \(C_{R}\) is a contour of radius \(R\) around the origin. In the limit, as \(R\) approaches infinity, the integral approaches zero. Using the residue theorem the integral can also be written as a sum of residues at the poles of the integrand. The poles are at zero, the positive and negative integers. The sum of the residues yields precisely twice the desired summation plus the residue at zero. Because the integral approaches zero, the sum of all the residues must be zero. The summation must therefore equal minus one half times the residue at zero. From the series expansion of the cotangent function \[ \cot(x)=\frac{1}{x} - \frac {x}{3} - \frac {x^3} {45} +\ldots, \] we see that the residue at zero is \(-\pi^{4}/45\) which yields the desired result.

 

 

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