리만 가설

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

• 디리클레 L-함수

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

재미있는 사실

• 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

• 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