"데데킨트 합"의 두 판 사이의 차이

2021년 2월 17일 (수) 05:02 기준 최신판

개요

데데킨트 에타함수의 모듈라 성질을 기술하기 위하여 도입
코탄젠트 합으로 쓸 수 있다

정의

다음과 같이 sawtooth 함수를 정의하자\[\left((x)\right)= \begin{cases} x-\lfloor x\rfloor - 1/2 & \mbox{ if }x\in\mathbb{R}\setminus\mathbb{Z} \\ 0 & \mbox{ if } x\in\mathbb{Z} \end{cases}\]

여기서 \(\lfloor x\rfloor\)는 \(x\)이하의 최대정수함수 (가우스함수)

그래프는 다음과 같다

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\(h,k\,(k>0)\)에 대하여 다음 등식이 성립함

\[ s(h,k)=\frac{1}{4k}\sum_{n=1}^{k-1} \cot \left( \frac{\pi n}{k} \right) \cot \left( \frac{\pi nh}{k} \right) \]

상호법칙

(정리) 데데킨트 서로 소인 양의 정수 \(c\)와 \(d\)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

\(p\geq 1\)이고, \(q,r\)은 \(p\)와 서로 소인 정수

\[ \begin{align} S(p;q,r)&=\sum_{k=1}^{p-1}\left( \left( \frac{qk}{p} \right) \right) \left( \left( \frac{rk}{p} \right) \right) \\ &=\frac{1}{4p}\sum_{k=1}^{p-1} \cot \left( \frac{\pi qk}{p} \right) \cot \left( \frac{\pi rk}{p} \right) \end{align} \]

라데마커 상호법칙 : 서로 소인 정수 \(p,q,r\geq 1\)에 대하여, 다음이 성립한다

\[ S(p;q,r)+S(q;r,p)+S(r;p,q)=\frac{ \left(p^2+q^2+r^2-3 p q r\right)}{12 p q r} \]

마르코프 수

h,k가 작은 경우 데데킨트합의 목록

\[ \begin{array}{c|c|c|c|c|c} h & k & s(h,k) & s(k,h) & s(h,k)+s(k,h) & \frac{1}{12}\left(\frac{k}{h}+\frac{h}{k}+\frac{1}{h k}\right)-\frac{1}{4} \\ \hline 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 2 & 0 & 0 & 0 & 0 \\ 1 & 3 & \frac{1}{18} & 0 & \frac{1}{18} & \frac{1}{18} \\ 2 & 3 & -\frac{1}{18} & 0 & -\frac{1}{18} & -\frac{1}{18} \\ 1 & 4 & \frac{1}{8} & 0 & \frac{1}{8} & \frac{1}{8} \\ 3 & 4 & -\frac{1}{8} & \frac{1}{18} & -\frac{5}{72} & -\frac{5}{72} \\ 1 & 5 & \frac{1}{5} & 0 & \frac{1}{5} & \frac{1}{5} \\ 2 & 5 & 0 & 0 & 0 & 0 \\ 3 & 5 & 0 & -\frac{1}{18} & -\frac{1}{18} & -\frac{1}{18} \\ 4 & 5 & -\frac{1}{5} & \frac{1}{8} & -\frac{3}{40} & -\frac{3}{40} \\ 1 & 6 & \frac{5}{18} & 0 & \frac{5}{18} & \frac{5}{18} \\ 5 & 6 & -\frac{5}{18} & \frac{1}{5} & -\frac{7}{90} & -\frac{7}{90} \\ 1 & 7 & \frac{5}{14} & 0 & \frac{5}{14} & \frac{5}{14} \\ 2 & 7 & \frac{1}{14} & 0 & \frac{1}{14} & \frac{1}{14} \\ 3 & 7 & -\frac{1}{14} & \frac{1}{18} & -\frac{1}{63} & -\frac{1}{63} \\ 4 & 7 & \frac{1}{14} & -\frac{1}{8} & -\frac{3}{56} & -\frac{3}{56} \\ 5 & 7 & -\frac{1}{14} & 0 & -\frac{1}{14} & -\frac{1}{14} \\ 6 & 7 & -\frac{5}{14} & \frac{5}{18} & -\frac{5}{63} & -\frac{5}{63} \\ 1 & 8 & \frac{7}{16} & 0 & \frac{7}{16} & \frac{7}{16} \\ 3 & 8 & \frac{1}{16} & -\frac{1}{18} & \frac{1}{144} & \frac{1}{144} \\ 5 & 8 & -\frac{1}{16} & 0 & -\frac{1}{16} & -\frac{1}{16} \\ 7 & 8 & -\frac{7}{16} & \frac{5}{14} & -\frac{9}{112} & -\frac{9}{112} \\ 1 & 9 & \frac{14}{27} & 0 & \frac{14}{27} & \frac{14}{27} \\ 2 & 9 & \frac{4}{27} & 0 & \frac{4}{27} & \frac{4}{27} \\ 4 & 9 & -\frac{4}{27} & \frac{1}{8} & -\frac{5}{216} & -\frac{5}{216} \\ 5 & 9 & \frac{4}{27} & -\frac{1}{5} & -\frac{7}{135} & -\frac{7}{135} \\ 7 & 9 & -\frac{4}{27} & \frac{1}{14} & -\frac{29}{378} & -\frac{29}{378} \\ 8 & 9 & -\frac{14}{27} & \frac{7}{16} & -\frac{35}{432} & -\frac{35}{432} \\ 1 & 10 & \frac{3}{5} & 0 & \frac{3}{5} & \frac{3}{5} \\ 3 & 10 & 0 & \frac{1}{18} & \frac{1}{18} & \frac{1}{18} \\ 7 & 10 & 0 & -\frac{1}{14} & -\frac{1}{14} & -\frac{1}{14} \\ 9 & 10 & -\frac{3}{5} & \frac{14}{27} & -\frac{11}{135} & -\frac{11}{135} \\ \end{array} \]

역사

수학사 연표

매스매티카 파일 및 계산 리소스

사전 형태의 자료

리뷰, 에세이, 강의노트

Grosswald, Emil. ‘Dedekind-Rademacher Sums’. The American Mathematical Monthly 78, no. 6 (1 June 1971): 639–44. doi:10.2307/2316571.

메타데이터

위키데이터

ID : Q2463775

Spacy 패턴 목록

[{'LOWER': 'dedekind'}, {'LEMMA': 'sum'}]

@@ 2번째 줄: / 2번째 줄: @@
 * [[데데킨트 에타함수]]의 모듈라 성질을 기술하기 위하여 도입
+* 코탄젠트 합으로 쓸 수 있다
@@ 21번째 줄: / 21번째 줄: @@
 ==코탄젠트합으로서의 표현==
+*  서로 소인 두 정수<math>h,k\,(k>0)</math>에 대하여 다음 등식이 성립함
-*  서로 소인 두 정수<math>b,c\,(c>0)</math>에 대하여 다음 등식이 성립함:<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>
+:<math>
+s(h,k)=\frac{1}{4k}\sum_{n=1}^{k-1}  \cot \left( \frac{\pi n}{k} \right) \cot \left( \frac{\pi nh}{k} \right)
+</math>
@@ 30번째 줄: / 32번째 줄: @@
 ==상호법칙==
-*  (정리) 데데킨트 서로 소인 양의 정수 <math>d</math>와 <math>c</math>에 대하여 다음이 성립한다.:<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>
+*  (정리) 데데킨트 서로 소인 양의 정수 <math>c</math>와 <math>d</math>에 대하여 다음이 성립한다
+:<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>
@@ 73번째 줄: / 76번째 줄: @@
 ==일반화==
+* <math>p\geq 1</math>이고, <math>q,r</math>은 <math>p</math>와 서로 소인 정수
-<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>
+\begin{align}
+S(p;q,r)&=\sum_{k=1}^{p-1}\left( \left( \frac{qk}{p} \right) \right)  \left( \left( \frac{rk}{p} \right) \right) \\
+&=\frac{1}{4p}\sum_{k=1}^{p-1}  \cot \left( \frac{\pi qk}{p} \right) \cot \left( \frac{\pi rk}{p} \right)
+\end{align}
+</math>
+* 라데마커 상호법칙 : 서로 소인 정수 <math>p,q,r\geq 1</math>에 대하여, 다음이 성립한다
+:<math>
+S(p;q,r)+S(q;r,p)+S(r;p,q)=\frac{ \left(p^2+q^2+r^2-3 p q r\right)}{12 p q r}
+</math>
+* [[마르코프 수]]
+==h,k가 작은 경우 데데킨트합의 목록==
+:<math>
-==h,k가 작은 경우 데데킨트합의 목록==
+\begin{array}{c|c|c|c|c|c}
+ h & k & s(h,k) & s(k,h) & s(h,k)+s(k,h) & \frac{1}{12}\left(\frac{k}{h}+\frac{h}{k}+\frac{1}{h k}\right)-\frac{1}{4} \\
-* <math>s(h,k)</math> 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)
+\hline
+& 1 & 0 & 0 & 0 & 0 \\
+& 2 & 0 & 0 & 0 & 0 \\
+& 3 & \frac{1}{18} & 0 & \frac{1}{18} & \frac{1}{18} \\
+& 3 & -\frac{1}{18} & 0 & -\frac{1}{18} & -\frac{1}{18} \\
+& 4 & \frac{1}{8} & 0 & \frac{1}{8} & \frac{1}{8} \\
+& 4 & -\frac{1}{8} & \frac{1}{18} & -\frac{5}{72} & -\frac{5}{72} \\
+& 5 & \frac{1}{5} & 0 & \frac{1}{5} & \frac{1}{5} \\
+& 5 & 0 & 0 & 0 & 0 \\
+& 5 & 0 & -\frac{1}{18} & -\frac{1}{18} & -\frac{1}{18} \\
+& 5 & -\frac{1}{5} & \frac{1}{8} & -\frac{3}{40} & -\frac{3}{40} \\
+& 6 & \frac{5}{18} & 0 & \frac{5}{18} & \frac{5}{18} \\
+& 6 & -\frac{5}{18} & \frac{1}{5} & -\frac{7}{90} & -\frac{7}{90} \\
+& 7 & \frac{5}{14} & 0 & \frac{5}{14} & \frac{5}{14} \\
+& 7 & \frac{1}{14} & 0 & \frac{1}{14} & \frac{1}{14} \\
+& 7 & -\frac{1}{14} & \frac{1}{18} & -\frac{1}{63} & -\frac{1}{63} \\
+& 7 & \frac{1}{14} & -\frac{1}{8} & -\frac{3}{56} & -\frac{3}{56} \\
+& 7 & -\frac{1}{14} & 0 & -\frac{1}{14} & -\frac{1}{14} \\
+& 7 & -\frac{5}{14} & \frac{5}{18} & -\frac{5}{63} & -\frac{5}{63} \\
+& 8 & \frac{7}{16} & 0 & \frac{7}{16} & \frac{7}{16} \\
+& 8 & \frac{1}{16} & -\frac{1}{18} & \frac{1}{144} & \frac{1}{144} \\
+& 8 & -\frac{1}{16} & 0 & -\frac{1}{16} & -\frac{1}{16} \\
+& 8 & -\frac{7}{16} & \frac{5}{14} & -\frac{9}{112} & -\frac{9}{112} \\
+& 9 & \frac{14}{27} & 0 & \frac{14}{27} & \frac{14}{27} \\
+& 9 & \frac{4}{27} & 0 & \frac{4}{27} & \frac{4}{27} \\
+& 9 & -\frac{4}{27} & \frac{1}{8} & -\frac{5}{216} & -\frac{5}{216} \\
+& 9 & \frac{4}{27} & -\frac{1}{5} & -\frac{7}{135} & -\frac{7}{135} \\
+& 9 & -\frac{4}{27} & \frac{1}{14} & -\frac{29}{378} & -\frac{29}{378} \\
+& 9 & -\frac{14}{27} & \frac{7}{16} & -\frac{35}{432} & -\frac{35}{432} \\
+& 10 & \frac{3}{5} & 0 & \frac{3}{5} & \frac{3}{5} \\
+& 10 & 0 & \frac{1}{18} & \frac{1}{18} & \frac{1}{18} \\
+& 10 & 0 & -\frac{1}{14} & -\frac{1}{14} & -\frac{1}{14} \\
+& 10 & -\frac{3}{5} & \frac{14}{27} & -\frac{11}{135} & -\frac{11}{135} \\
+\end{array}
+</math>
 ==역사==
@@ 120번째 줄: / 164번째 줄: @@
 * http://ko.wikipedia.org/wiki/
 * http://en.wikipedia.org/wiki/Dedekind_sum
-* http://en.wikipedia.org/wiki/
 * http://mathworld.wolfram.com/DedekindSum.html
 ==관련도서==
-* [http://math.sfsu.edu/beck/ccd.html Computing the Continuous Discretely: Integer-Point Enumeration in Polyhedra]
+* Matthias Beck and Sinai Robins [http://math.sfsu.edu/beck/ccd.html Computing the Continuous Discretely: Integer-Point Enumeration in Polyhedra], Springer, 2007
-**  Matthias Beck and Sinai Robins, Springer, 2007
+* H. Rademacher and E. Grosswald, Dedekind Sums, The Carus Mathematical Monographs
-*  Dedekind Sums, The Carus Mathematical Monographs
-**  H. Rademacher and E. Grosswald
+==리뷰, 에세이, 강의노트==
+* Grosswald, Emil. ‘Dedekind-Rademacher Sums’. The American Mathematical Monthly 78, no. 6 (1 June 1971): 639–44. doi:10.2307/2316571.
 ==관련논문==
+* Genki Shibukawa, New trigonometric identities and reciprocity laws of generalized Dedekind sums, http://arxiv.org/abs/1409.2451v4
+* Rassias, Michael Th, and László Tóth. “Trigonometric Representations of Generalized Dedekind and Hardy Sums via the Discrete Fourier Transform.” arXiv:1512.01466 [math], December 4, 2015. http://arxiv.org/abs/1512.01466.
+* Rassias, Michael Th. “A Cotangent Sum Related to Zeros of the Estermann Zeta Function.” arXiv:1512.04711 [math], December 15, 2015. http://arxiv.org/abs/1512.04711.
+* Burrin, Claire. “Generalized Dedekind Sums and Equidistribution Mod 1.” arXiv:1509.04429 [math], September 15, 2015. http://arxiv.org/abs/1509.04429.
+* Dowker, J. S. “On Sums of Powers of Cosecs.” arXiv:1507.01848 [hep-Th], July 7, 2015. http://arxiv.org/abs/1507.01848.
+* Tsukerman, Emmanuel. “A Generalization of Zolotarev’s Lemma and Equality of Dedekind Sums Mod <math>8 \mathbb{Z}</math>.” arXiv:1501.03544 [math], January 14, 2015. http://arxiv.org/abs/1501.03544.
+* Maier, Helmut, and Michael Th Rassias. “The Order of Magnitude for Moments for Certain Cotangent Sums.” arXiv:1412.1512 [math], December 3, 2014. http://arxiv.org/abs/1412.1512.
+* Shibukawa, Genki. “New Trigonometric Identities and Reciprocity Laws of Generalized Dedekind Sums.” arXiv:1409.2451 [math], September 8, 2014. http://arxiv.org/abs/1409.2451.
+* Beck, Matthias. ‘Dedekind Cotangent Sums’. arXiv:math/0112077, 7 December 2001. http://arxiv.org/abs/math/0112077.
+* Gunnells, Paul E., and Robert Sczech. ‘Evaluation of Dedekind Sums, Eisenstein Cocycles, and Special Values of L-Functions’. Duke Mathematical Journal 118, no. 2 (1 June 2003): 229–60. doi:10.1215/S0012-7094-03-11822-0.
+* Asai, Tetsuya. ‘The Reciprocity of Dedekind Sums and the Factor Set for the Universal Covering Group of <math>{\rm SL}(2,\,R)</math>’. Nagoya Mathematical Journal 37 (1970): 67–80. http://projecteuclid.org/euclid.nmj/1118797877
-* [http://arxiv.org/abs/math.NT/0112077 Dedekind cotangent sums]
-** Matthias Beck, Acta Arithmetica 109, no.2 (2003), 109-130
-* [http://www.jstor.org/stable/2316571?&Search=yes&term=Emil&term=Grosswald,&term=,&term=%22Dedekind-Rademacher+sums%22&list=hide&searchUri=/action/doBasicSearch%3FQuery%3DEmil%2BGrosswald%252C%2B%2522%2BDedekind-Rademacher%2Bsums%2B%2522%252C%26x%3D0%26y%3D0%26wc%3Don&item=1&ttl=3&returnArticleService=showArticle Dedekind-Rademacher Sums]
-** Emil Grosswald, The American Mathematical Monthly, Vol. 78, No. 6 (Jun. - Jul., 1971), pp. 639-644
-* [http://projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.nmj/1118797877&page=record The reciprocity of Dedekind sums and the factor set for the universal covering group of] <math>{\rm SL}(2,\,R)</math>
-** Tetsuya Asai, Source: Nagoya Math. J. Volume 37 (1970), 67-80.
+[[분류:정수론]]
-[[분류:정수론]]
+==메타데이터==
+===위키데이터===
+* ID :  [https://www.wikidata.org/wiki/Q2463775 Q2463775]
+===Spacy 패턴 목록===
+* [{'LOWER': 'dedekind'}, {'LEMMA': 'sum'}]