데데킨트 합

간단한 소개

• 다음과 같이 sawtooth 함수를 정의하자
  \(\left((x)\right)= \begin{cases} x-\lfloor x\rfloor - 1/2 & \mbox{ if }x\in\mathbb{R}\setminus\mathbb{Z} \\ 0 & \mbox{ if } x\in\mathbb{Z} \end{cases}\)
  \(\lfloor x\rfloor\)는 
  
• 예
  \(((0.8))=0.8-0-0.5=0.3\)
  \(((-0.2))=-0.2-(-1)-0.5=0.3\)
  

• 서로 소인 두 정수 \(h, k>0\)에 대하여 데데킨트 합 \(s(h,k)\)은 다음과 같이 정의됨
  \(s(h,k)=\sum_{n\mod k} \left( \left( \frac{n}{k} \right) \right) \left( \left( \frac{hn}{k} \right) \right)\)
  

 

• 서로 소인 두 정수 \(b,c>0\)에 대하여 다음 등식이 성립함
  \(s(b,c)=\frac{1}{4c}\sum_{n=1}^{c-1} \cot \left( \frac{\pi n}{c} \right) \cot \left( \frac{\pi nb}{c} \right)\)
  

 

 

상호법칙

(정리) 데데킨트
서로 소인 양의 정수 \(d\)와 \(c\)에 대하여 다음이 성립한다.

\(s(d,c)+s(c,d) =\frac{1}{12}\left(\frac{d}{c}+\frac{1}{dc}+\frac{c}{d}\right)-\frac{1}{4}\)

 

 

증명

\(F(z)=\cot \pi z\, \cot \pi cz\, \cot \pi dz\)

사각형 \(\pm iM, 1+\pm iM\) 을 조금 수정하여 0은 포함하고, 1은 빠지도록 하는 컨투어 \(\Gamma\)에 대한 적분을 사용한다.

\(\lim_{M\to \infty}\cot (x+iM)=-i\)이므로, \(\lim_{M\to \infty}F(x+iM)=-i\) 임을 확인하자.

\(\int_{\Gamma}F(z)dz\) 는 \(M\)에 의존하지 않으므로, \(\int_{\Gamma}F(z)dz = \lim_{M\to\infty}\int_{\Gamma}F(z)dz=-2i\)을 얻는다.

따라서 \(\Gamma\) 내부에 있는 유수의 합 \(S\)는 \(-\frac{1}{\pi}\) 가 된다.

 

폴은 다음과 같은 점에서 발생한다.

• \(z=0\)
  
• \(z=\lambda/c\,, \lambda=1,2,\cdots, c-1\)
  
• \(z=\mu/d\,, \mu=1,2,\cdots, d-1\)
  

\(z=\lambda/c\) 에서의 유수는 \(\frac{1}{\pi c}\cot \frac{\pi \lambda}{c}\cot\frac{\pi d\lambda}{c}\)

\(z=\mu/c\) 에서의 유수는 \(\frac{1}{\pi d}\cot \frac{\pi \mu}{c}\cot\frac{\pi d\mu}{c}\)

 

 코탄젠트의 급수전개를 사용하여 \(z=0\)에서의 유수를 구하자.

\(F(z)=\cot \pi z\, \cot \pi cz\, \cot \pi dz =\frac{1}{\pi^3 cd z^3}(1-\frac{\pi^2z^2}{3}-\cdots)(1-\frac{\pi^2z^2d^2}{3}-\cdots)(1-\frac{\pi^2z^2c^2}{3}-\cdots)\)

따라서 \(z=0\)에서의 유수는 \(-\frac{1}{3\pi}\left(\frac{d}{c}+\frac{1}{cd}+\frac{c}{d}\right)\) 이다. 

 

\(S=\frac{4}{\pi}[-\frac{1}{12}\left(\frac{d}{c}+\frac{1}{dc}+\frac{c}{d}\right)+s(d,c)+s(c,d)]=-\frac{1}{\pi}\) 를 얻는다. (증명끝)

 

 

일반화

\(D(a,b;c)=\sum_{n\mod c} \left( \left( \frac{an}{c} \right) \right) \left( \left( \frac{bn}{c} \right) \right)\)

 

 

b,c가 작은 경우 데데킨트합의 목록

s(0,0)=Null
s(0,1)=0
s(0,2)=Null
s(0,3)=Null
s(0,4)=Null
s(0,5)=Null
s(0,6)=Null
s(0,7)=Null
s(0,8)=Null
s(0,9)=Null
s(0,10)=Null
s(1,0)=0
s(1,1)=0
s(1,2)=0
s(1,3)=1/18
s(1,4)=1/8
s(1,5)=1/5
s(1,6)=5/18
s(1,7)=5/14
s(1,8)=7/16
s(1,9)=14/27
s(1,10)=3/5
s(2,0)=Null
s(2,1)=0
s(2,2)=Null
s(2,3)=-(1/18)
s(2,4)=Null
s(2,5)=0
s(2,6)=Null
s(2,7)=1/14
s(2,8)=Null
s(2,9)=4/27
s(2,10)=Null
s(3,0)=Null
s(3,1)=0
s(3,2)=0
s(3,3)=Null
s(3,4)=-(1/8)
s(3,5)=0
s(3,6)=Null
s(3,7)=-(1/14)
s(3,8)=1/16
s(3,9)=Null
s(3,10)=0
s(4,0)=Null
s(4,1)=0
s(4,2)=Null
s(4,3)=1/18
s(4,4)=Null
s(4,5)=-(1/5)
s(4,6)=Null
s(4,7)=1/14
s(4,8)=Null
s(4,9)=-(4/27)
s(4,10)=Null
s(5,0)=Null
s(5,1)=0
s(5,2)=0
s(5,3)=-(1/18)
s(5,4)=1/8
s(5,5)=Null
s(5,6)=-(5/18)
s(5,7)=-(1/14)
s(5,8)=-(1/16)
s(5,9)=4/27
s(5,10)=Null
s(6,0)=Null
s(6,1)=0
s(6,2)=Null
s(6,3)=Null
s(6,4)=Null
s(6,5)=1/5
s(6,6)=Null
s(6,7)=-(5/14)
s(6,8)=Null
s(6,9)=Null
s(6,10)=Null
s(7,0)=Null
s(7,1)=0
s(7,2)=0
s(7,3)=1/18
s(7,4)=-(1/8)
s(7,5)=0
s(7,6)=5/18
s(7,7)=Null
s(7,8)=-(7/16)
s(7,9)=-(4/27)
s(7,10)=0
s(8,0)=Null
s(8,1)=0
s(8,2)=Null
s(8,3)=-(1/18)
s(8,4)=Null
s(8,5)=0
s(8,6)=Null
s(8,7)=5/14
s(8,8)=Null
s(8,9)=-(14/27)
s(8,10)=Null
s(9,0)=Null
s(9,1)=0
s(9,2)=0
s(9,3)=Null
s(9,4)=1/8
s(9,5)=-(1/5)
s(9,6)=Null
s(9,7)=1/14
s(9,8)=7/16
s(9,9)=Null
s(9,10)=-(3/5)
s(10,0)=Null
s(10,1)=0
s(10,2)=Null
s(10,3)=1/18
s(10,4)=Null
s(10,5)=Null
s(10,6)=Null
s(10,7)=-(1/14)
s(10,8)=Null
s(10,9)=14/27
s(10,10)=Null

 

 

 

 

역사

• 수학사연표
  

 

 

관련된 다른 주제들

• 데데킨트 에타함수
  
• 코탄젠트
  

 

 

수학용어번역

• http://www.google.com/dictionary?langpair=en%7Cko&q=
• 대한수학회 수학 학술 용어집
  
  • http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=eng_term&fstr=
• 대한수학회 수학용어한글화 게시판

 

 

사전 형태의 자료

• http://ko.wikipedia.org/wiki/
• http://en.wikipedia.org/wiki/Dedekind_sum
• http://en.wikipedia.org/wiki/
• http://mathworld.wolfram.com/DedekindSum.html
• http://www.wolframalpha.com/input/?i=
• NIST Digital Library of Mathematical Functions
• The On-Line Encyclopedia of Integer Sequences
  
  • http://www.research.att.com/~njas/sequences/?q=

 

 

 

관련도서 및 추천도서

• Computing the Continuous Discretely: Integer-Point Enumeration in Polyhedra
  
  • Matthias Beck and Sinai Robins, Springer, 2007
    
• Dedekind Sums, The Carus Mathematical Monographs
  
  • H. Rademacher and E. Grosswald
    
• 도서내검색
  
  • http://books.google.com/books?q=
  • http://book.daum.net/search/contentSearch.do?query=
• 도서검색
  
  • http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=dedekind+sums
  • http://book.daum.net/search/mainSearch.do?query=

 

 

관련논문

• Dedekind cotangent sums
  
  • Matthias Beck, Acta Arithmetica 109, no.2 (2003), 109-130   
•  
  
• Dedekind-Rademacher Sums
  
  • Emil Grosswald, The American Mathematical Monthly, Vol. 78, No. 6 (Jun. - Jul., 1971), pp. 639-644
• The reciprocity of Dedekind sums and the factor set for the universal covering group of \({\rm SL}(2,\,R)\)
  
  • Tetsuya Asai, Source: Nagoya Math. J. Volume 37 (1970), 67-80.
• http://ko.wikipedia.org/wiki/
• http://en.wikipedia.org/wiki/Dedekind_sum
• http://www.wolframalpha.com/input/?i=sawtooth+function
   
  

 

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