"리만 가설"의 두 판 사이의 차이

수학노트
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==개요==
  
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*  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>
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*  자명한 해는 <math>s=-2,-4,-6\cdots</math>
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*  리만제타함수의 자명하지 않은 해(비자명해)는 그 실수부가 <math>1/2</math> 이라는 추측
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==소수정리==
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*  리만 제타 함수와 소수 계량 함수의 관계
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*  "모든 실수 t에 대하여 <math>\zeta(1+it)\neq 0 </math> 이다" 는 소수정리와 동치명제이다
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* [[소수정리]]
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==비자명해의 수론적 특성==
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*  추측
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**  The positive imaginary parts of nontrivial zeros of <math>\zeta(s)</math> are linearly independent over <math>\mathbb{Q}</math>
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==일반화된 리만가설==
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* [[디리클레 L-함수]]
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==응용==
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*  Rubinstein-Sarnak 1994
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** how often <math>\pi(x)>\operatorname{Li}(x)</math>
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*  even(x) : number of natural numbers , even number of prime factors
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*  Odd(x) : odd number of prime factors
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* [[골드바흐 추측]]
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*  1923 하디-리틀우드
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*  1937비노그라도프
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*  1997 Deshouillers-Effinger-te Riele-Zinoviev
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* [[순환소수에 대한 아틴의 추측]]
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:<math>C_{\mathrm{Artin}}=\prod_{q\ \mathrm{prime}} \left(1-\frac{1}{q(q-1)}\right) = 0.3739558136\ldots.</math>
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* 1967 Hooley
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* 1973 Weinberger
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* [[이차 수체 유클리드 도메인의 분류]]
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==Spectal theory and RH==
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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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==Hilbert-Polya and random matrices==
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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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* 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.
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* 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.
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* 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.
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*  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
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* 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.
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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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* Paul Bourgade, Marc Yor. [http://hal.archives-ouvertes.fr/hal-00119410/en/ Random Matrices and the Riemann zeta function], 2006
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* http://www.secamlocal.ex.ac.uk/people/staff/mrwatkin/zeta/random.htm
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* http://www.maths.bris.ac.uk/~majpk/publications.html
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==Noncommutatative geometry==
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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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==Computation of non-trivial zeros==
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* R. P. Brent, “On the zeros of the Riemann zeta function in the critical strip”, Mathematics of Computation 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/~odlyzko/doc/arch/fast.zeta.eval.pdf
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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/~brent/pd/rpb047.pdf
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==메모==
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* 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.
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* 영화속 오류 russell crowe riemann zeta
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* http://mathoverflow.net/questions/13647/why-does-the-riemann-zeta-function-have-non-trivial-zeros
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==역사==
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* [[수학사 연표]]
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==관련된 항목들==
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* [[소수정리]]
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* [[리만제타함수의 영점]]
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* [[클레이 연구소 밀레니엄 문제들]]
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==사전 형태의 자료==
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* http://ko.wikipedia.org/wiki/리만가설
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* http://en.wikipedia.org/wiki/Riemann_hypothesis
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* http://en.wikipedia.org/wiki/Riemann-Siegel_formula
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* 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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==리뷰, 에세이, 강의노트==
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* Alain, Connes. “An Essay on the Riemann Hypothesis.” arXiv:1509.05576 [math], September 18, 2015. http://arxiv.org/abs/1509.05576.
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* Wolf, Marek. “Will a Physicists Prove the Riemann Hypothesis?” arXiv:1410.1214 [math-Ph], October 5, 2014. http://arxiv.org/abs/1410.1214.
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* 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.
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* http://logic.pdmi.ras.ru/~yumat/talks/turku_rh_2014/index.php
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* 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
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* Bombieri, E. "Problems of the millennium: The Riemann hypothesis." Clay mathematical Institute (2000).
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==관련논문==
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* Brian Conrey, Jonathan P. Keating, Moments of zeta and correlations of divisor-sums: IV, http://arxiv.org/abs/1603.06893v1
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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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[[분류:리만 제타 함수]]
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==메타데이터==
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===위키데이터===
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* ID :  [https://www.wikidata.org/wiki/Q2993323 Q2993323]
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===Spacy 패턴 목록===
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* [{'LOWER': 'hilbert'}, {'OP': '*'}, {'LOWER': 'pólya'}, {'LEMMA': 'conjecture'}]

2021년 2월 17일 (수) 04:41 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

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



Spectal theory and RH



Hilbert-Polya and random matrices

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


메모

역사



관련된 항목들


사전 형태의 자료


리뷰, 에세이, 강의노트

관련논문

메타데이터

위키데이터

Spacy 패턴 목록

  • [{'LOWER': 'hilbert'}, {'OP': '*'}, {'LOWER': 'pólya'}, {'LEMMA': 'conjecture'}]
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