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* http://en.wikipedia.org/wiki/Hilbert-P%C3%B3lya_conjecture | * http://en.wikipedia.org/wiki/Hilbert-P%C3%B3lya_conjecture | ||
* http://en.wikipedia.org/wiki/Gaussian_Unitary_Ensemble | * http://en.wikipedia.org/wiki/Gaussian_Unitary_Ensemble | ||
| − | * | + | * 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. |
| + | * 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, | * 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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==Noncommutatative geometry== | ==Noncommutatative geometry== | ||
2015년 1월 1일 (목) 19:12 판
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
- http://en.wikipedia.org/wiki/Hilbert-P%C3%B3lya_conjecture
- http://en.wikipedia.org/wiki/Gaussian_Unitary_Ensemble
- 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.
- 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,
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
- http://hal.archives-ouvertes.fr/hal-00119410/en/
- http://www.secamlocal.ex.ac.uk/people/staff/mrwatkin/zeta/random.htm
- Random Matrices and the Riemann zeta function
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
재미있는 사실
- 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
- 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
- Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse Bernhard Riemann, November 1859