리만가설

이 항목의 스프링노트 원문주소

리만가설

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 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)>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
이차수체 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
Introduction to the Random Matrix Theory: Gaussian Unitary Ensemble and Beyond Authors: Yan V. Fyodorov
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

수학사연표

관련된 항목들

감마함수

수학용어번역

대한수학회 수학 학술 용어집
- http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=eng_term&fstr=
대한수학회 수학용어한글화 게시판

사전 형태의 자료

관련논문

Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse Bernhard Riemann, November 1859