데데킨트 합

 

 

 

개요

• 데데킨트 에타함수의 모듈라 성질을 기술하기 위하여 도입
  

 

 

정의

• 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 \(x\)이하의 최대정수함수(가우스함수)
  
• 예
  \(((0.8))=0.8-0-0.5=0.3\)
  \(((-0.2))=-0.2-(-1)-0.5=0.3\)
  
• 서로 소인 두 정수\(h, k\,(k>0)\)에 대하여 데데킨트 합 \(s(h,k)\)은 다음과 같이 정의됨
  \(s(h,k)=\sum_{n\mod k} \left( \left( \frac{n}{k} \right) \right) \left( \left( \frac{hn}{k} \right) \right)\)
  \(s(h,k)=\sum_{n=1}^{k-1} \frac{n}{k} \left( \left( \frac{hn}{k} \right) \right)\)
   
  

 

 

코탄젠트합으로서의 표현

• 서로 소인 두 정수\(b,c\,(c>0)\)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. ■

 

 

일반화

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

 

 

h,k가 작은 경우 데데킨트합의 목록

• \(s(h,k)\)
  s(1,1)=0
  s(1,2)=0
  s(1,3)=1/18
  s(2,3)=-(1/18)
  s(1,4)=1/8
  s(3,4)=-(1/8)
  s(1,5)=1/5
  s(2,5)=0
  s(3,5)=0
  s(4,5)=-(1/5)
  s(1,6)=5/18
  s(5,6)=-(5/18)
  s(1,7)=5/14
  s(2,7)=1/14
  s(3,7)=-(1/14)
  s(4,7)=1/14
  s(5,7)=-(1/14)
  s(6,7)=-(5/14)
  s(1,8)=7/16
  s(3,8)=1/16
  s(5,8)=-(1/16)
  s(7,8)=-(7/16)
  s(1,9)=14/27
  s(2,9)=4/27
  s(4,9)=-(4/27)
  s(5,9)=4/27
  s(7,9)=-(4/27)
  s(8,9)=-(14/27)
  s(1,10)=3/5
  s(3,10)=0
  s(7,10)=0
  s(9,10)=-(3/5)
  s(1,11)=15/22
  s(2,11)=5/22
  s(3,11)=3/22
  s(4,11)=3/22
  s(5,11)=-(5/22)
  s(6,11)=5/22
  s(7,11)=-(3/22)
  s(8,11)=-(3/22)
  s(9,11)=-(5/22)
  s(10,11)=-(15/22)
  s(1,12)=55/72
  s(5,12)=-(1/72)
  s(7,12)=1/72
  s(11,12)=-(55/72)
  s(1,13)=11/13
  s(2,13)=4/13
  s(3,13)=1/13
  s(4,13)=-(1/13)
  s(5,13)=0
  s(6,13)=-(4/13)
  s(7,13)=4/13
  s(8,13)=0
  s(9,13)=1/13
  s(10,13)=-(1/13)
  s(11,13)=-(4/13)
  s(12,13)=-(11/13)
  s(1,14)=13/14
  s(3,14)=3/14
  s(5,14)=3/14
  s(9,14)=-(3/14)
  s(11,14)=-(3/14)
  s(13,14)=-(13/14)
  s(1,15)=91/90
  s(2,15)=7/18
  s(4,15)=19/90
  s(7,15)=-(7/18)
  s(8,15)=7/18
  s(11,15)=-(19/90)
  s(13,15)=-(7/18)
  s(14,15)=-(91/90)
  s(1,16)=35/32
  s(3,16)=5/32
  s(5,16)=-(5/32)
  s(7,16)=-(3/32)
  s(9,16)=3/32
  s(11,16)=5/32
  s(13,16)=-(5/32)
  s(15,16)=-(35/32)
  s(1,17)=20/17
  s(2,17)=8/17
  s(3,17)=5/17
  s(4,17)=0
  s(5,17)=1/17
  s(6,17)=5/17
  s(7,17)=1/17
  s(8,17)=-(8/17)
  s(9,17)=8/17
  s(10,17)=-(1/17)
  s(11,17)=-(5/17)
  s(12,17)=-(1/17)
  s(13,17)=0
  s(14,17)=-(5/17)
  s(15,17)=-(8/17)
  s(16,17)=-(20/17)
  s(1,18)=34/27
  s(5,18)=2/27
  s(7,18)=-(2/27)
  s(11,18)=2/27
  s(13,18)=-(2/27)
  s(17,18)=-(34/27)
  s(1,19)=51/38
  s(2,19)=21/38
  s(3,19)=9/38
  s(4,19)=11/38
  s(5,19)=11/38
  s(6,19)=-(9/38)
  s(7,19)=3/38
  s(8,19)=-(3/38)
  s(9,19)=-(21/38)
  s(10,19)=21/38
  s(11,19)=3/38
  s(12,19)=-(3/38)
  s(13,19)=9/38
  s(14,19)=-(11/38)
  s(15,19)=-(11/38)
  s(16,19)=-(9/38)
  s(17,19)=-(21/38)
  s(18,19)=-(51/38)
  s(1,20)=57/40
  s(3,20)=3/8
  s(7,20)=3/8
  s(9,20)=-(7/40)
  s(11,20)=7/40
  s(13,20)=-(3/8)
  s(17,20)=-(3/8)
  s(19,20)=-(57/40)
  

 

 

 

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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