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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>
  
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* [[리만가설]]
  
 
 
 
 
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[[리만가설]]
 
[[리만가설]]
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<h5>Spectal theory and RH</h5>
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* [http://projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.dmj/1077311789&page=record The Selberg trace formula and the Riemann zeta function]
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* Dennis A. Hejhal
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<h5>Hilbert-Polya</h5>
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* http://en.wikipedia.org/wiki/Hilbert-P%C3%B3lya_conjecture
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* http://en.wikipedia.org/wiki/Gaussian_Unitary_Ensemble
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* [http://arxiv.org/abs/math-ph/0412017 Introduction to the Random Matrix Theory: Gaussian Unitary Ensemble and Beyond] Authors: Yan V. Fyodorov
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* Montgomery, Hugh L. (1973), "The pair correlation of zeros of the zeta function", Analytic number theory, Proc. Sympos. Pure Math., XXIV, Providence, R.I.: American Mathematical Society, pp. 181–193,
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<h5>Noncommutatative geometry</h5>
  
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영화속 오류 russell crowe riemann zeta
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* Noncommutative Geometry, Quantum Fields, and Motives Alain Connes, Matilde Marcolli
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* [http://gigapedia.com/items:description?id=90484 Noncommutative Geometry and Number Theory]: Where Arithmetic Meets Geometry and Physics (Aspects of Mathematics) Caterina Consani, Matilde Marcolli (Eds.)
  
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http://mathoverflow.net/questions/13647/why-does-the-riemann-zeta-function-have-non-trivial-zeros
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<h5 style="line-height: 3.42em; margin: 0px; font-family: 'malgun gothic',dotum,gulim,sans-serif; background-position: 0px 100%; color: rgb(34, 61, 103); font-size: 1.16em;">역사</h5>
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<h5>Random matrices</h5>
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* http://hal.archives-ouvertes.fr/hal-00119410/en/
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* http://www.secamlocal.ex.ac.uk/people/staff/mrwatkin/zeta/random.htm
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* Random Matrices and the Riemann zeta function
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* [http://people.reed.edu/%7Ejerry/311/zeta.pdf http://people.reed.edu/~jerry/311/zeta.pdf] analytic continuation''
  
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* [[수학사연표 (역사)|수학사연표]]
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<h5>Computation of non-trivial zeros</h5>
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R. P. Brent, “On the zeros of the Riemann zeta function in the critical strip”, Mathematics of Computation<br> 33 (1979), 1361–1372.
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* http://numbers.computation.free.fr/Constants/Miscellaneous/zetaevaluations.pdf
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The Riemann-Siegel Expansion for the Zeta Function: High Orders and Remainders, M. V. Berry
  
 
 
 
 
  
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[http://www.dtc.umn.edu/%7Eodlyzko/doc/arch/fast.zeta.eval.pdf http://www.dtc.umn.edu/~odlyzko/doc/arch/fast.zeta.eval.pdf]
  
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* [[감마함수]]<br>
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[http://www.mathematik.hu-berlin.de/%7Egaggle/EVENTS/2006/BRENT60/presentations/Herman%20J.J.%20te%20Riele%20-%20Separation%20of%20the%20complex%20zeros%20of%20the%20Riemann%20zeta%20function.pdf http://www.mathematik.hu-berlin.de/~gaggle/EVENTS/2006/BRENT60/presentations/Herman%20J.J.%20te%20Riele%20-%20Separation%20of%20the%20complex%20zeros%20of%20the%20Riemann%20zeta%20function.pdf]
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[http://wwwmaths.anu.edu.au/%7Ebrent/pd/rpb047.pdf http://wwwmaths.anu.edu.au/~brent/pd/rpb047.pdf]
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<h5 style="line-height: 3.42em; margin: 0px; font-family: 'malgun gothic',dotum,gulim,sans-serif; background-position: 0px 100%; color: rgb(34, 61, 103); font-size: 1.16em;">수학용어번역</h5>
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*  영화속 오류 russell crowe riemann zeta<br>
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* http://mathoverflow.net/questions/13647/why-does-the-riemann-zeta-function-have-non-trivial-zeros<br>
  
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* [http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=&fstr= 대한수학회 수학 학술 용어집]<br>
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** http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=eng_term&fstr=
 
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* [http://kms.or.kr/home/kor/board/bulletin_list_subject.asp?bulletinid=%7BD6048897-56F9-43D7-8BB6-50B362D1243A%7D&boardname=%BC%F6%C7%D0%BF%EB%BE%EE%C5%E4%B7%D0%B9%E6&globalmenu=7&localmenu=4 대한수학회 수학용어한글화 게시판]
 
  
 
 
 
 
  
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* [http://ko.wikipedia.org/wiki/%EB%A6%AC%EB%A7%8C%EA%B0%80%EC%84%A4 http://ko.wikipedia.org/wiki/리만가설]
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* [[수학사연표 (역사)|수학사연표]]
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* http://en.wikipedia.org/wiki/Riemann_hypothesis
 
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* http://www.wolframalpha.com/input/?i=Riemann+zeta
 
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* [http://dlmf.nist.gov/ NIST Digital Library of Mathematical Functions]
 
  
 
 
 
 
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* [http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Zeta/ Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse]<br>
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* [[감마함수]]<br>
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** [http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Zeta/ ]Bernhard Riemann, November 1859
 
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* http://www.jstor.org/action/doBasicSearch?Query=
 
  
 
 
 
 
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* 도서내검색<br>
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* [http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=&fstr= 대한수학회 수학 학술 용어집]<br>
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** http://books.google.com/books?q=
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** http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=eng_term&fstr=
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** http://book.daum.net/search/contentSearch.do?query=
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* [http://kms.or.kr/home/kor/board/bulletin_list_subject.asp?bulletinid=%7BD6048897-56F9-43D7-8BB6-50B362D1243A%7D&boardname=%BC%F6%C7%D0%BF%EB%BE%EE%C5%E4%B7%D0%B9%E6&globalmenu=7&localmenu=4 대한수학회 수학용어한글화 게시판]
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*  도서검색<br>
 
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** http://books.google.com/books?q=
 
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** http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=
 
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** http://book.daum.net/search/mainSearch.do?query=
 
  
 
 
 
 
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* 네이버 뉴스 검색 (키워드 수정)<br>
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* [http://ko.wikipedia.org/wiki/%EB%A6%AC%EB%A7%8C%EA%B0%80%EC%84%A4 http://ko.wikipedia.org/wiki/리만가설]
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** http://news.search.naver.com/search.naver?where=news&x=0&y=0&sm=tab_hty&query=
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* http://en.wikipedia.org/wiki/Riemann_hypothesis
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** http://news.search.naver.com/search.naver?where=news&x=0&y=0&sm=tab_hty&query=
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* http://en.wikipedia.org/wiki/Riemann-Siegel_formula
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** http://news.search.naver.com/search.naver?where=news&x=0&y=0&sm=tab_hty&query=
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* [http://en.wikipedia.org/wiki/Riemann%E2%80%93Siegel_theta_function http://en.wikipedia.org/wiki/Riemann–Siegel_theta_function]
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* http://www.wolframalpha.com/input/?i=Riemann+zeta
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* [http://dlmf.nist.gov/ NIST Digital Library of Mathematical Functions]
  
 
 
 
 
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* 구글 블로그 검색 http://blogsearch.google.com/blogsearch?q=
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* [http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Zeta/ Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse] Bernhard Riemann, November 1859
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* [http://navercast.naver.com/science/list 네이버 오늘의과학]
 

2012년 4월 18일 (수) 16:47 판

이 항목의 스프링노트 원문주소

 

 

개요
  • 리만제타함수의 함수방정식은 다음과 같음
    \(\pi^{-s/2}\ \Gamma\left(\frac{s}{2}\right)\ \zeta(s)=\pi^{-(1-s)/2}\ \Gamma\left(\frac{1-s}{2}\right)\ \zeta(1-s)\)
  • 자명한 해는 \(s=-2,-4,-6\cdots\)
  • 리만제타함수의 자명하지 않은 해는 그 실수부가 \(1/2\) 이라는 추측

 

리만가설

리만가설

 

 

소수정리
  • "모든 실수 t에 대하여 \(\zeta(1+it)\neq 0 \) 이다" 는 소수정리와 동치명제이다
  • 소수정리

 

Conjecture

The positive imaginary parts of nontrivial zeros of \zeta(s) are linearly independent over \mathbb{Q}

 

 

일반화된 리만가설

 

 

 

응용

Rubinstein-Sarnak 1994

how often \pi(x)>Li(x)

 

 

even(x) : number of natural numbers , even number of prime factors

Odd(x) : odd number of prime factors

 

 

골드바흐 추측

1923 하디-리틀우드

1937비노그라도프

1997 Deshouillers-Effinger-te Riele-Zinoviev

 

 

순환소수에 대한 아틴의 추측

\(C_{\mathrm{Artin}}=\prod_{q\ \mathrm{prime}} \left(1-\frac{1}{q(q-1)}\right) = 0.3739558136\ldots.\)

1967 Hooley

 

이차수체 유클리드 도메인의 분류

1973 Weinberger

 

 

Spectal theory and RH

 

 

Hilbert-Polya

 

 

Noncommutatative geometry
  • Noncommutative Geometry, Quantum Fields, and Motives Alain Connes, Matilde Marcolli
  • Noncommutative Geometry and Number Theory: Where Arithmetic Meets Geometry and Physics (Aspects of Mathematics) Caterina Consani, Matilde Marcolli (Eds.)

 

 

Random matrices

 

 

Computation of non-trivial zeros

R. P. Brent, “On the zeros of the Riemann zeta function in the critical strip”, Mathematics of Computation
33 (1979), 1361–1372.

The Riemann-Siegel Expansion for the Zeta Function: High Orders and Remainders, M. V. Berry

 

http://www.dtc.umn.edu/~odlyzko/doc/arch/fast.zeta.eval.pdf

http://www.mathematik.hu-berlin.de/~gaggle/EVENTS/2006/BRENT60/presentations/Herman%20J.J.%20te%20Riele%20-%20Separation%20of%20the%20complex%20zeros%20of%20the%20Riemann%20zeta%20function.pdf

http://wwwmaths.anu.edu.au/~brent/pd/rpb047.pdf

 

 

재미있는 사실

 

 

역사

 

 

관련된 항목들

 

 

수학용어번역

 

 

사전 형태의 자료

 

 

관련논문
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