라마누잔의 class invariants

이 항목의 스프링노트 원문주소

• 라마누잔의 class invariants
  

 

 

간단한 소개

• 라마누잔이 많은 계산 결과를 남겨놓은 분야
  
• class field theory에서 중요한 역할을 함
  \(G_n:=2^{-1/4}f(\sqrt{-n})\)
  \(g_n:=2^{-1/4}f_1(\sqrt{-n})=2^{-1/4}\frac{\eta(\frac{\sqrt{-n}}{2})}{\eta(\sqrt{-n})}\)
  

 

 

필요한 정의

 

\(q=e^{2\pi i \tau}\)

• 자코비 세타함수
  [[자코비 세타함수|]]\(\theta_{2}(\tau)= \sum_{n=-\infty}^\infty q^{(n+\frac{1}{2})^2/2}\)
  \(\theta_3(\tau)=\sum_{n=-\infty}^\infty q^{n^2/2}\)
  \(\theta_{4}(\tau)= \sum_{n=-\infty}^\infty (-1)^n q^{n^2/2}\)
  
• 모듈라 군, j-invariant and the singular moduli
   
  [[모듈라 군, j-invariant and the singular moduli|]]
  \(k=k(\tau)=\frac{\theta_2^2(\tau)}{\theta_3^2(\tau)}\)
  \(k'=\sqrt{1-k^2}=\frac{\theta_4^2(\tau)}{\theta_3^2(\tau)}\)
  

 

• 데데킨트 에타함수
   \(\eta(\tau) = q^{1/24} \prod_{n=1}^{\infty} (1-q^{n})\)
  

• 베버(Weber) 모듈라 함수
  [[베버(Weber) 모듈라 함수|]]
  \(f(\tau)=\frac{e^{-\frac{\pi i}{24}}\eta(\frac{\tau+1}{2})}{\eta(\tau)}=q^{-1/48} \prod_{n=1}^{\infty} (1+q^{n-\frac{1}{2}})\)
  \(f_1(\tau)=\frac{\eta(\frac{\tau}{2})}{\eta(\tau)}=q^{-1/48} \prod_{n=1}^{\infty} (1-q^{n-\frac{1}{2}})\)
  \(f_2(\tau)=\sqrt{2}\frac{\eta(2\tau)}{\eta(\tau)}=\sqrt{2}q^{1/24} \prod_{n=1}^{\infty} (1+q^{n})\)
  

 

 

 

special values

\(G_{25}=\frac{\sqrt{5}+1}{2}\)

\(g_{10}=\sqrt{\frac{\sqrt{5}+1}{2}}\)

 

\(g_{58}^2=\frac{\sqrt{29}+5}{2}\)

 

class invariants의 계산

• 크로네커 극한 공식의 이용
  

 

 

 

메모

 

\(G_n:=(2kk')^{-1/12}=2^{-1/4}f(\sqrt{-n})\)

\(g_n:=(\frac{k'(\sqrt{-n})^2}{2k(\sqrt{-n})})^{1/12}=2^{-1/4}f_1(\sqrt{-n})\)

 

 

 

 

 

 

관련된 다른 주제들

• 오일러의 convenient number ( Idoneal number)
• 라마누잔과 파이
•  

 

 

사전형태의 참고자료

• http://ko.wikipedia.org/wiki/
• http://en.wikipedia.org/wiki/
• http://scholar.google.com/scholar?q=ramanujan%27s+class+invariants&hl=ko&lr=&start=10&sa=N
• http://front.math.ucdavis.edu/search?a=&t=&c=&n=40&s=Listings&q=
• http://www.ams.org/mathscinet/search/publications.html?pg4=AUCN&s4=&co4=AND&pg5=TI&s5=&co5=AND&pg6=PC&s6=&co6=AND&pg7=ALLF&co7=AND&Submit=Search&dr=all&yrop=eq&arg3=&yearRangeFirst=&yearRangeSecond=&pg8=ET&s8=All&s7=

 

관련도서 및 추천도서

• Ramanujan's Notebooks: V
  
  • Bruce C. Berndt
• 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=
  • http://book.daum.net/search/mainSearch.do?query=

 

 

관련논문과 에세이

• Ramanujan's Most Singular Modulus
  
  • Mark B. Villarino, Arxiv, 2003-8

• Ramanujan and the modular j-invariant
  
  • BC Berndt, HH Chan, Canadian Mathematical Bulletin, 1999

• RAMANUJAN–WEBER CLASS INVARIANT Gn AND WATSON'S EMPIRICAL PROCESS
  
  • Heng Huat Chan, Journal of the London Mathematical Society, 1998
• Ramanujan's Class Invariants With Applications To The Values Of q-Continued Fractions And Theta-Functions
  
  • Bruce C. Berndt ,  Heng Huat Chan ,  Liang-Cheng Zhang, 1997

• Ramanujan's class invariants, Kronecker's limit formula, and modular equations
  
  • Bruce C. Berndt ,  Heng Huat Chan ,  Liang-Cheng Zhang Transactions of the American Mathematical Society, 1997
• Ramanujan’s class invariants and cubic continued fraction
  
  • Bruce C. Berndt ,  Heng Huat Chan ,  Liang-Cheng Zhang, ACTA ARITHMETICA LXXIII.1 (1995)

• Some singular moduli (II)
• G.N. Watson, Q J Math 1932 os-3: 189-212

• Some singular moduli (I)
  
  • G.N. Watson

 

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