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이 항목의 스프링노트 원문주소==
Pythagoras0 (토론 | 기여) 잔글 (찾아 바꾸기 – “<h5>” 문자열을 “==” 문자열로) |
Pythagoras0 (토론 | 기여) 잔글 (찾아 바꾸기 – “</h5>” 문자열을 “==” 문자열로) |
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| − | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">이 항목의 스프링노트 원문주소 | + | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">이 항목의 스프링노트 원문주소== |
* [[데데킨트 합]] | * [[데데킨트 합]] | ||
| 7번째 줄: | 7번째 줄: | ||
| − | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">개요 | + | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">개요== |
* [[데데킨트 에타함수]]의 모듈라 성질을 기술하기 위하여 도입<br> | * [[데데킨트 에타함수]]의 모듈라 성질을 기술하기 위하여 도입<br> | ||
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| − | <h5 style="margin: 0px; line-height: 2em;">정의 | + | <h5 style="margin: 0px; line-height: 2em;">정의== |
* 다음과 같이 sawtooth 함수를 정의하자<br><math>\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}</math><br><math>\lfloor x\rfloor</math>는 <math>x</math>이하의 [[최대정수함수 (가우스함수)]]<br> | * 다음과 같이 sawtooth 함수를 정의하자<br><math>\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}</math><br><math>\lfloor x\rfloor</math>는 <math>x</math>이하의 [[최대정수함수 (가우스함수)]]<br> | ||
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| − | <h5 style="margin: 0px; line-height: 2em;">코탄젠트합으로서의 표현 | + | <h5 style="margin: 0px; line-height: 2em;">코탄젠트합으로서의 표현== |
* 서로 소인 두 정수<math>b,c\,(c>0)</math>에 대하여 다음 등식이 성립함<br><math>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)</math><br> | * 서로 소인 두 정수<math>b,c\,(c>0)</math>에 대하여 다음 등식이 성립함<br><math>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)</math><br> | ||
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| − | <h5 style="margin: 0px; line-height: 2em;">상호법칙 | + | <h5 style="margin: 0px; line-height: 2em;">상호법칙== |
* (정리) 데데킨트<br> 서로 소인 양의 정수 <math>d</math>와 <math>c</math>에 대하여 다음이 성립한다.<br><math>s(d,c)+s(c,d) =\frac{1}{12}\left(\frac{d}{c}+\frac{1}{dc}+\frac{c}{d}\right)-\frac{1}{4}</math><br> | * (정리) 데데킨트<br> 서로 소인 양의 정수 <math>d</math>와 <math>c</math>에 대하여 다음이 성립한다.<br><math>s(d,c)+s(c,d) =\frac{1}{12}\left(\frac{d}{c}+\frac{1}{dc}+\frac{c}{d}\right)-\frac{1}{4}</math><br> | ||
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| − | <h5 style="margin: 0px; line-height: 2em;">일반화 | + | <h5 style="margin: 0px; line-height: 2em;">일반화== |
<math>D(a,b;c)=\sum_{n\mod c} \left( \left( \frac{an}{c} \right) \right) \left( \left( \frac{bn}{c} \right) \right)</math> | <math>D(a,b;c)=\sum_{n\mod c} \left( \left( \frac{an}{c} \right) \right) \left( \left( \frac{bn}{c} \right) \right)</math> | ||
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| − | <h5 style="margin: 0px; line-height: 2em;">h,k가 작은 경우 데데킨트합의 목록 | + | <h5 style="margin: 0px; line-height: 2em;">h,k가 작은 경우 데데킨트합의 목록== |
* <math>s(h,k)</math><br> s(1,1)=0<br> s(1,2)=0<br> s(1,3)=1/18<br> s(2,3)=-(1/18)<br> s(1,4)=1/8<br> s(3,4)=-(1/8)<br> s(1,5)=1/5<br> s(2,5)=0<br> s(3,5)=0<br> s(4,5)=-(1/5)<br> s(1,6)=5/18<br> s(5,6)=-(5/18)<br> s(1,7)=5/14<br> s(2,7)=1/14<br> s(3,7)=-(1/14)<br> s(4,7)=1/14<br> s(5,7)=-(1/14)<br> s(6,7)=-(5/14)<br> s(1,8)=7/16<br> s(3,8)=1/16<br> s(5,8)=-(1/16)<br> s(7,8)=-(7/16)<br> s(1,9)=14/27<br> s(2,9)=4/27<br> s(4,9)=-(4/27)<br> s(5,9)=4/27<br> s(7,9)=-(4/27)<br> s(8,9)=-(14/27)<br> s(1,10)=3/5<br> s(3,10)=0<br> s(7,10)=0<br> s(9,10)=-(3/5)<br> s(1,11)=15/22<br> s(2,11)=5/22<br> s(3,11)=3/22<br> s(4,11)=3/22<br> s(5,11)=-(5/22)<br> s(6,11)=5/22<br> s(7,11)=-(3/22)<br> s(8,11)=-(3/22)<br> s(9,11)=-(5/22)<br> s(10,11)=-(15/22)<br> s(1,12)=55/72<br> s(5,12)=-(1/72)<br> s(7,12)=1/72<br> s(11,12)=-(55/72)<br> s(1,13)=11/13<br> s(2,13)=4/13<br> s(3,13)=1/13<br> s(4,13)=-(1/13)<br> s(5,13)=0<br> s(6,13)=-(4/13)<br> s(7,13)=4/13<br> s(8,13)=0<br> s(9,13)=1/13<br> s(10,13)=-(1/13)<br> s(11,13)=-(4/13)<br> s(12,13)=-(11/13)<br> s(1,14)=13/14<br> s(3,14)=3/14<br> s(5,14)=3/14<br> s(9,14)=-(3/14)<br> s(11,14)=-(3/14)<br> s(13,14)=-(13/14)<br> s(1,15)=91/90<br> s(2,15)=7/18<br> s(4,15)=19/90<br> s(7,15)=-(7/18)<br> s(8,15)=7/18<br> s(11,15)=-(19/90)<br> s(13,15)=-(7/18)<br> s(14,15)=-(91/90)<br> s(1,16)=35/32<br> s(3,16)=5/32<br> s(5,16)=-(5/32)<br> s(7,16)=-(3/32)<br> s(9,16)=3/32<br> s(11,16)=5/32<br> s(13,16)=-(5/32)<br> s(15,16)=-(35/32)<br> s(1,17)=20/17<br> s(2,17)=8/17<br> s(3,17)=5/17<br> s(4,17)=0<br> s(5,17)=1/17<br> s(6,17)=5/17<br> s(7,17)=1/17<br> s(8,17)=-(8/17)<br> s(9,17)=8/17<br> s(10,17)=-(1/17)<br> s(11,17)=-(5/17)<br> s(12,17)=-(1/17)<br> s(13,17)=0<br> s(14,17)=-(5/17)<br> s(15,17)=-(8/17)<br> s(16,17)=-(20/17)<br> s(1,18)=34/27<br> s(5,18)=2/27<br> s(7,18)=-(2/27)<br> s(11,18)=2/27<br> s(13,18)=-(2/27)<br> s(17,18)=-(34/27)<br> s(1,19)=51/38<br> s(2,19)=21/38<br> s(3,19)=9/38<br> s(4,19)=11/38<br> s(5,19)=11/38<br> s(6,19)=-(9/38)<br> s(7,19)=3/38<br> s(8,19)=-(3/38)<br> s(9,19)=-(21/38)<br> s(10,19)=21/38<br> s(11,19)=3/38<br> s(12,19)=-(3/38)<br> s(13,19)=9/38<br> s(14,19)=-(11/38)<br> s(15,19)=-(11/38)<br> s(16,19)=-(9/38)<br> s(17,19)=-(21/38)<br> s(18,19)=-(51/38)<br> s(1,20)=57/40<br> s(3,20)=3/8<br> s(7,20)=3/8<br> s(9,20)=-(7/40)<br> s(11,20)=7/40<br> s(13,20)=-(3/8)<br> s(17,20)=-(3/8)<br> s(19,20)=-(57/40)<br> | * <math>s(h,k)</math><br> s(1,1)=0<br> s(1,2)=0<br> s(1,3)=1/18<br> s(2,3)=-(1/18)<br> s(1,4)=1/8<br> s(3,4)=-(1/8)<br> s(1,5)=1/5<br> s(2,5)=0<br> s(3,5)=0<br> s(4,5)=-(1/5)<br> s(1,6)=5/18<br> s(5,6)=-(5/18)<br> s(1,7)=5/14<br> s(2,7)=1/14<br> s(3,7)=-(1/14)<br> s(4,7)=1/14<br> s(5,7)=-(1/14)<br> s(6,7)=-(5/14)<br> s(1,8)=7/16<br> s(3,8)=1/16<br> s(5,8)=-(1/16)<br> s(7,8)=-(7/16)<br> s(1,9)=14/27<br> s(2,9)=4/27<br> s(4,9)=-(4/27)<br> s(5,9)=4/27<br> s(7,9)=-(4/27)<br> s(8,9)=-(14/27)<br> s(1,10)=3/5<br> s(3,10)=0<br> s(7,10)=0<br> s(9,10)=-(3/5)<br> s(1,11)=15/22<br> s(2,11)=5/22<br> s(3,11)=3/22<br> s(4,11)=3/22<br> s(5,11)=-(5/22)<br> s(6,11)=5/22<br> s(7,11)=-(3/22)<br> s(8,11)=-(3/22)<br> s(9,11)=-(5/22)<br> s(10,11)=-(15/22)<br> s(1,12)=55/72<br> s(5,12)=-(1/72)<br> s(7,12)=1/72<br> s(11,12)=-(55/72)<br> s(1,13)=11/13<br> s(2,13)=4/13<br> s(3,13)=1/13<br> s(4,13)=-(1/13)<br> s(5,13)=0<br> s(6,13)=-(4/13)<br> s(7,13)=4/13<br> s(8,13)=0<br> s(9,13)=1/13<br> s(10,13)=-(1/13)<br> s(11,13)=-(4/13)<br> s(12,13)=-(11/13)<br> s(1,14)=13/14<br> s(3,14)=3/14<br> s(5,14)=3/14<br> s(9,14)=-(3/14)<br> s(11,14)=-(3/14)<br> s(13,14)=-(13/14)<br> s(1,15)=91/90<br> s(2,15)=7/18<br> s(4,15)=19/90<br> s(7,15)=-(7/18)<br> s(8,15)=7/18<br> s(11,15)=-(19/90)<br> s(13,15)=-(7/18)<br> s(14,15)=-(91/90)<br> s(1,16)=35/32<br> s(3,16)=5/32<br> s(5,16)=-(5/32)<br> s(7,16)=-(3/32)<br> s(9,16)=3/32<br> s(11,16)=5/32<br> s(13,16)=-(5/32)<br> s(15,16)=-(35/32)<br> s(1,17)=20/17<br> s(2,17)=8/17<br> s(3,17)=5/17<br> s(4,17)=0<br> s(5,17)=1/17<br> s(6,17)=5/17<br> s(7,17)=1/17<br> s(8,17)=-(8/17)<br> s(9,17)=8/17<br> s(10,17)=-(1/17)<br> s(11,17)=-(5/17)<br> s(12,17)=-(1/17)<br> s(13,17)=0<br> s(14,17)=-(5/17)<br> s(15,17)=-(8/17)<br> s(16,17)=-(20/17)<br> s(1,18)=34/27<br> s(5,18)=2/27<br> s(7,18)=-(2/27)<br> s(11,18)=2/27<br> s(13,18)=-(2/27)<br> s(17,18)=-(34/27)<br> s(1,19)=51/38<br> s(2,19)=21/38<br> s(3,19)=9/38<br> s(4,19)=11/38<br> s(5,19)=11/38<br> s(6,19)=-(9/38)<br> s(7,19)=3/38<br> s(8,19)=-(3/38)<br> s(9,19)=-(21/38)<br> s(10,19)=21/38<br> s(11,19)=3/38<br> s(12,19)=-(3/38)<br> s(13,19)=9/38<br> s(14,19)=-(11/38)<br> s(15,19)=-(11/38)<br> s(16,19)=-(9/38)<br> s(17,19)=-(21/38)<br> s(18,19)=-(51/38)<br> s(1,20)=57/40<br> s(3,20)=3/8<br> s(7,20)=3/8<br> s(9,20)=-(7/40)<br> s(11,20)=7/40<br> s(13,20)=-(3/8)<br> s(17,20)=-(3/8)<br> s(19,20)=-(57/40)<br> | ||
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| − | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">역사 | + | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">역사== |
* [[수학사연표 (역사)|수학사연표]]<br> | * [[수학사연표 (역사)|수학사연표]]<br> | ||
| 103번째 줄: | 103번째 줄: | ||
| − | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">관련된 다른 주제들 | + | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">관련된 다른 주제들== |
* [[데데킨트 에타함수]]<br> | * [[데데킨트 에타함수]]<br> | ||
| 113번째 줄: | 113번째 줄: | ||
| − | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">수학용어번역 | + | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">수학용어번역== |
* http://www.google.com/dictionary?langpair=en|ko&q= | * http://www.google.com/dictionary?langpair=en|ko&q= | ||
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| − | ==매스매티카 파일 및 계산 리소스[[3985465/attachments/4981790|3985465/attachments/4981790]] | + | ==매스매티카 파일 및 계산 리소스[[3985465/attachments/4981790|3985465/attachments/4981790]]== |
* https://docs.google.com/leaf?id=0B8XXo8Tve1cxNmU3MzVlOTctOTIxYi00ZDZkLTkwMzgtYjJkZTY5Y2Q1NDkw&sort=name&layout=list&num=50 | * https://docs.google.com/leaf?id=0B8XXo8Tve1cxNmU3MzVlOTctOTIxYi00ZDZkLTkwMzgtYjJkZTY5Y2Q1NDkw&sort=name&layout=list&num=50 | ||
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| − | ==사전 형태의 자료 | + | ==사전 형태의 자료== |
* http://ko.wikipedia.org/wiki/ | * http://ko.wikipedia.org/wiki/ | ||
| 156번째 줄: | 156번째 줄: | ||
| − | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">관련도서 및 추천도서 | + | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">관련도서 및 추천도서== |
* [http://math.sfsu.edu/beck/ccd.html Computing the Continuous Discretely: Integer-Point Enumeration in Polyhedra]<br> | * [http://math.sfsu.edu/beck/ccd.html Computing the Continuous Discretely: Integer-Point Enumeration in Polyhedra]<br> | ||
| 173번째 줄: | 173번째 줄: | ||
| − | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">관련논문 | + | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">관련논문== |
* [http://arxiv.org/abs/math.NT/0112077 Dedekind cotangent sums]<br> | * [http://arxiv.org/abs/math.NT/0112077 Dedekind cotangent sums]<br> | ||
| 187번째 줄: | 187번째 줄: | ||
| − | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">블로그 | + | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">블로그== |
* 구글 블로그 검색 http://blogsearch.google.com/blogsearch?q= | * 구글 블로그 검색 http://blogsearch.google.com/blogsearch?q= | ||
* 네이버 블로그 검색 http://cafeblog.search.naver.com/search.naver?where=post&sm=tab_jum&query= | * 네이버 블로그 검색 http://cafeblog.search.naver.com/search.naver?where=post&sm=tab_jum&query= | ||
2012년 11월 1일 (목) 13:14 판
이 항목의 스프링노트 원문주소==
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)
\(F(z)=\cot \pi z\, \cot \pi cz\, \cot \pi dz\)
네 점 \(\pm iM, 1+\pm iM\)을 꼭지점으로 갖는 사각형을 조금 수정하여 0은 포함하고, 1은 빠지도록 하는 폐곡선 \(\Gamma\)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 \(\frac{1}{\pi d}\cot \frac{\pi \mu}{d}\cot\frac{\pi c\mu}{d}\)
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==
관련된 다른 주제들==
수학용어번역==
매스매티카 파일 및 계산 리소스3985465/attachments/4981790
- https://docs.google.com/leaf?id=0B8XXo8Tve1cxNmU3MzVlOTctOTIxYi00ZDZkLTkwMzgtYjJkZTY5Y2Q1NDkw&sort=name&layout=list&num=50
- http://www.wolframalpha.com/input/?i=
- http://functions.wolfram.com/
- NIST Digital Library of Mathematical Functions
- The On-Line Encyclopedia of Integer Sequences
- Numbers, constants and computation
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
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
- 도서검색
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
블로그==
- 데데킨트 에타함수의 모듈라 성질을 기술하기 위하여 도입
- 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
\(s(h,k)=\sum_{n\mod k} \left( \left( \frac{n}{k} \right) \right) \left( \left( \frac{hn}{k} \right) \right)\)
\(s(h,k)=\sum_{n=1}^{k-1} \frac{n}{k} \left( \left( \frac{hn}{k} \right) \right)\)
- 서로 소인 두 정수\(b,c\,(c>0)\)에 대하여 다음 등식이 성립함
\(s(b,c)=\frac{1}{4c}\sum_{n=1}^{c-1} \cot \left( \frac{\pi n}{c} \right) \cot \left( \frac{\pi nb}{c} \right)\)
- (정리) 데데킨트
서로 소인 양의 정수 \(d\)와 \(c\)에 대하여 다음이 성립한다.
\(s(d,c)+s(c,d) =\frac{1}{12}\left(\frac{d}{c}+\frac{1}{dc}+\frac{c}{d}\right)-\frac{1}{4}\)
- \(z=0\)
- \(z=\lambda/c\,, \lambda=1,2,\cdots, c-1\)
- \(z=\mu/d\,, \mu=1,2,\cdots, d-1\)
- \(s(h,k)\)
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)
- Computing the Continuous Discretely: Integer-Point Enumeration in Polyhedra
- Matthias Beck and Sinai Robins, Springer, 2007
- Matthias Beck and Sinai Robins, Springer, 2007
- Dedekind Sums, The Carus Mathematical Monographs
- H. Rademacher and E. Grosswald
- H. Rademacher and E. Grosswald
- 도서내검색
- 도서검색
- 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