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Pythagoras0 (토론 | 기여) |
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| (사용자 2명의 중간 판 36개는 보이지 않습니다) | |||
| 1번째 줄: | 1번째 줄: | ||
| − | + | ==개요== | |
| − | + | * 리만제타함수의 함수방정식은 다음과 같음:<math>\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)</math> | |
| + | * 자명한 해는 <math>s=-2,-4,-6\cdots</math> | ||
| + | * 리만제타함수의 자명하지 않은 해(비자명해)는 그 실수부가 <math>1/2</math> 이라는 추측 | ||
| − | + | ||
| − | + | ||
| − | + | ==소수정리== | |
| − | * | + | * 리만 제타 함수와 소수 계량 함수의 관계 |
| − | |||
| − | + | * "모든 실수 t에 대하여 <math>\zeta(1+it)\neq 0 </math> 이다" 는 소수정리와 동치명제이다 | |
| + | * [[소수정리]] | ||
| − | + | ||
| − | + | ||
| − | + | ==비자명해의 수론적 특성== | |
| − | < | + | * 추측 |
| + | ** The positive imaginary parts of nontrivial zeros of <math>\zeta(s)</math> are linearly independent over <math>\mathbb{Q}</math> | ||
| − | + | ||
| − | + | ||
| − | + | ==일반화된 리만가설== | |
| − | + | * [[디리클레 L-함수]] | |
| − | + | ||
| − | + | ||
| − | + | ||
| − | |||
| − | |||
| − | + | ==응용== | |
| − | < | + | * Rubinstein-Sarnak 1994 |
| + | ** how often <math>\pi(x)>\operatorname{Li}(x)</math> | ||
| + | * 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]] | ||
| + | :<math>C_{\mathrm{Artin}}=\prod_{q\ \mathrm{prime}} \left(1-\frac{1}{q(q-1)}\right) = 0.3739558136\ldots.</math> | ||
| + | * 1967 Hooley | ||
| + | * 1973 Weinberger | ||
| + | * [[이차 수체 유클리드 도메인의 분류]] | ||
| − | * [http:// | + | |
| − | * http:// | + | |
| − | * http://www. | + | |
| − | * [http:// | + | |
| + | ==Spectal theory and RH== | ||
| + | |||
| + | * [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] | ||
| + | * Dennis A. Hejhal | ||
| + | |||
| + | |||
| + | |||
| + | |||
| + | |||
| + | ==Hilbert-Polya and random matrices== | ||
| + | |||
| + | * http://en.wikipedia.org/wiki/Hilbert-P%C3%B3lya_conjecture | ||
| + | * http://en.wikipedia.org/wiki/Gaussian_Unitary_Ensemble | ||
| + | * Forrester, Peter J., and Anthony Mays. “Finite Size Corrections in Random Matrix Theory and Odlyzko’s Data Set for the Riemann Zeros.” arXiv:1506.06531 [math-Ph], June 22, 2015. http://arxiv.org/abs/1506.06531. | ||
| + | * Barrett, Owen, Frank W. K. Firk, Steven J. Miller, and Caroline Turnage-Butterbaugh. ‘From Quantum Systems to L-Functions: Pair Correlation Statistics and Beyond’. arXiv:1505.07481 [math], 27 May 2015. http://arxiv.org/abs/1505.07481. | ||
| + | * Fyodorov, Yan V. “Introduction to the Random Matrix Theory: Gaussian Unitary Ensemble and Beyond.” arXiv:math-ph/0412017, December 7, 2004. http://arxiv.org/abs/math-ph/0412017. | ||
| + | * J.P. Keating (1993), Quantum chaology and the Riemann zeta-function, in Quantum Chaos, eds. G. Casati, I. Guarneri & U. Smilansky, (North-Holland, Amsterdam), 145-185 http://www.maths.bris.ac.uk/~majpk/papers/13.pdf | ||
| + | * Berry, M. V. “The Bakerian Lecture, 1987: Quantum Chaology.” Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 413, no. 1844 (September 8, 1987): 183–98. doi:10.1098/rspa.1987.0109. | ||
| + | * 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, | ||
| + | * Paul Bourgade, Marc Yor. [http://hal.archives-ouvertes.fr/hal-00119410/en/ Random Matrices and the Riemann zeta function], 2006 | ||
| + | * http://www.secamlocal.ex.ac.uk/people/staff/mrwatkin/zeta/random.htm | ||
| + | * http://www.maths.bris.ac.uk/~majpk/publications.html | ||
| + | |||
| + | ==Noncommutatative geometry== | ||
| + | |||
| + | * Noncommutative Geometry, Quantum Fields, and Motives Alain Connes, Matilde Marcolli | ||
| + | * [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.) | ||
| + | |||
| + | |||
| − | + | ==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. | ||
| + | * http://numbers.computation.free.fr/Constants/Miscellaneous/zetaevaluations.pdf | ||
| + | * 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/%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] | ||
| + | * http://wwwmaths.anu.edu.au/~brent/pd/rpb047.pdf | ||
| − | + | ||
| − | + | ==메모== | |
| + | * Feng, Nianrong, Yongzheng Wang, and Ruixin Wu. “To Reveal the Truth of the Zeta Function in Riemann’s Manuscript.” arXiv:1508.02932 [math], August 6, 2015. http://arxiv.org/abs/1508.02932. | ||
| + | * 영화속 오류 russell crowe riemann zeta | ||
| + | * http://mathoverflow.net/questions/13647/why-does-the-riemann-zeta-function-have-non-trivial-zeros | ||
| − | + | ==역사== | |
| − | |||
| − | |||
| − | + | * [[수학사 연표]] | |
| − | + | ||
| − | + | ||
| − | + | ==관련된 항목들== | |
| − | + | * [[소수정리]] | |
| − | + | * [[리만제타함수의 영점]] | |
| − | * | + | * [[클레이 연구소 밀레니엄 문제들]] |
| − | * | + | |
| − | * | ||
| − | |||
| − | + | ||
| + | ==사전 형태의 자료== | ||
| − | + | * http://ko.wikipedia.org/wiki/리만가설 | |
| + | * http://en.wikipedia.org/wiki/Riemann_hypothesis | ||
| + | * http://en.wikipedia.org/wiki/Riemann-Siegel_formula | ||
| + | * http://en.wikipedia.org/wiki/Riemann–Siegel_theta_function | ||
| + | * http://www.wolframalpha.com/input/?i=Riemann+zeta | ||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | + | ==리뷰, 에세이, 강의노트== | |
| + | * Alain, Connes. “An Essay on the Riemann Hypothesis.” arXiv:1509.05576 [math], September 18, 2015. http://arxiv.org/abs/1509.05576. | ||
| + | * Wolf, Marek. “Will a Physicists Prove the Riemann Hypothesis?” arXiv:1410.1214 [math-Ph], October 5, 2014. http://arxiv.org/abs/1410.1214. | ||
| + | * França, Guilherme, and André LeClair. 2014. “A Theory for the Zeros of Riemann Zeta and Other L-Functions.” arXiv:1407.4358 [hep-Th, Physics:math-Ph], July. http://arxiv.org/abs/1407.4358. | ||
| + | * http://logic.pdmi.ras.ru/~yumat/talks/turku_rh_2014/index.php | ||
| + | * Conrey, J. Brian. "The Riemann hypothesis." Notices of the AMS 50, no. 3 (2003): 341-353. http://www.ams.org/notices/200303/fea-conrey-web.pdf | ||
| + | * Bombieri, E. "Problems of the millennium: The Riemann hypothesis." Clay mathematical Institute (2000). | ||
| − | + | ==관련논문== | |
| + | * Brian Conrey, Jonathan P. Keating, Moments of zeta and correlations of divisor-sums: IV, http://arxiv.org/abs/1603.06893v1 | ||
| + | * [http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Zeta/ Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse] Bernhard Riemann, November 1859 | ||
| − | + | [[분류:리만 제타 함수]] | |
| − | * | + | ==메타데이터== |
| − | * [ | + | ===위키데이터=== |
| + | * ID : [https://www.wikidata.org/wiki/Q2993323 Q2993323] | ||
| + | ===Spacy 패턴 목록=== | ||
| + | * [{'LOWER': 'hilbert'}, {'OP': '*'}, {'LOWER': 'pólya'}, {'LEMMA': 'conjecture'}] | ||
2021년 2월 17일 (수) 05:41 기준 최신판
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 해(비자명해)는 그 실수부가 \(1/2\) 이라는 추측
소수정리
- 리만 제타 함수와 소수 계량 함수의 관계
- "모든 실수 t에 대하여 \(\zeta(1+it)\neq 0 \) 이다" 는 소수정리와 동치명제이다
- 소수정리
비자명해의 수론적 특성
- 추측
- The positive imaginary parts of nontrivial zeros of \(\zeta(s)\) are linearly independent over \(\mathbb{Q}\)
일반화된 리만가설
응용
- Rubinstein-Sarnak 1994
- how often \(\pi(x)>\operatorname{Li}(x)\)
- 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
- 1973 Weinberger
- 이차 수체 유클리드 도메인의 분류
Spectal theory and RH
- The Selberg trace formula and the Riemann zeta function
- Dennis A. Hejhal
Hilbert-Polya and random matrices
- http://en.wikipedia.org/wiki/Hilbert-P%C3%B3lya_conjecture
- http://en.wikipedia.org/wiki/Gaussian_Unitary_Ensemble
- Forrester, Peter J., and Anthony Mays. “Finite Size Corrections in Random Matrix Theory and Odlyzko’s Data Set for the Riemann Zeros.” arXiv:1506.06531 [math-Ph], June 22, 2015. http://arxiv.org/abs/1506.06531.
- Barrett, Owen, Frank W. K. Firk, Steven J. Miller, and Caroline Turnage-Butterbaugh. ‘From Quantum Systems to L-Functions: Pair Correlation Statistics and Beyond’. arXiv:1505.07481 [math], 27 May 2015. http://arxiv.org/abs/1505.07481.
- Fyodorov, Yan V. “Introduction to the Random Matrix Theory: Gaussian Unitary Ensemble and Beyond.” arXiv:math-ph/0412017, December 7, 2004. http://arxiv.org/abs/math-ph/0412017.
- J.P. Keating (1993), Quantum chaology and the Riemann zeta-function, in Quantum Chaos, eds. G. Casati, I. Guarneri & U. Smilansky, (North-Holland, Amsterdam), 145-185 http://www.maths.bris.ac.uk/~majpk/papers/13.pdf
- Berry, M. V. “The Bakerian Lecture, 1987: Quantum Chaology.” Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 413, no. 1844 (September 8, 1987): 183–98. doi:10.1098/rspa.1987.0109.
- 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,
- Paul Bourgade, Marc Yor. Random Matrices and the Riemann zeta function, 2006
- http://www.secamlocal.ex.ac.uk/people/staff/mrwatkin/zeta/random.htm
- http://www.maths.bris.ac.uk/~majpk/publications.html
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.)
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.
- http://numbers.computation.free.fr/Constants/Miscellaneous/zetaevaluations.pdf
- 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
메모
- Feng, Nianrong, Yongzheng Wang, and Ruixin Wu. “To Reveal the Truth of the Zeta Function in Riemann’s Manuscript.” arXiv:1508.02932 [math], August 6, 2015. http://arxiv.org/abs/1508.02932.
- 영화속 오류 russell crowe riemann zeta
- http://mathoverflow.net/questions/13647/why-does-the-riemann-zeta-function-have-non-trivial-zeros
역사
관련된 항목들
사전 형태의 자료
- http://ko.wikipedia.org/wiki/리만가설
- http://en.wikipedia.org/wiki/Riemann_hypothesis
- http://en.wikipedia.org/wiki/Riemann-Siegel_formula
- http://en.wikipedia.org/wiki/Riemann–Siegel_theta_function
- http://www.wolframalpha.com/input/?i=Riemann+zeta
리뷰, 에세이, 강의노트
- Alain, Connes. “An Essay on the Riemann Hypothesis.” arXiv:1509.05576 [math], September 18, 2015. http://arxiv.org/abs/1509.05576.
- Wolf, Marek. “Will a Physicists Prove the Riemann Hypothesis?” arXiv:1410.1214 [math-Ph], October 5, 2014. http://arxiv.org/abs/1410.1214.
- França, Guilherme, and André LeClair. 2014. “A Theory for the Zeros of Riemann Zeta and Other L-Functions.” arXiv:1407.4358 [hep-Th, Physics:math-Ph], July. http://arxiv.org/abs/1407.4358.
- http://logic.pdmi.ras.ru/~yumat/talks/turku_rh_2014/index.php
- Conrey, J. Brian. "The Riemann hypothesis." Notices of the AMS 50, no. 3 (2003): 341-353. http://www.ams.org/notices/200303/fea-conrey-web.pdf
- Bombieri, E. "Problems of the millennium: The Riemann hypothesis." Clay mathematical Institute (2000).
관련논문
- Brian Conrey, Jonathan P. Keating, Moments of zeta and correlations of divisor-sums: IV, http://arxiv.org/abs/1603.06893v1
- Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse Bernhard Riemann, November 1859
메타데이터
위키데이터
- ID : Q2993323
Spacy 패턴 목록
- [{'LOWER': 'hilbert'}, {'OP': '*'}, {'LOWER': 'pólya'}, {'LEMMA': 'conjecture'}]